Can we use the stored gravitational potential energy of a building to produce power? There are skyscrapers sitting and pushing on the ground with tremendous weight. Is it possible to convert this weight/force to harness energy to power the building? 
Maybe build the building on top of some type of pendulum that will rotate under the pressure, and when one cycle of rotation reaches the equilibrium point we could give it a kick from the stored energy to continue rotation.
Was something like this created or tested and found useless?
Note: maybe my question should be, is it possible to convert the potential energy of a building into kinetic?
 A: Of course you can get that stored potential energy out of the skyscraper.
Once
That's the whole problem. All the other answers dance around that. You can let the skyscraper drop somewhat and get out some energy, by some mechanism. But then you have a lowered skyscraper and all you got was a short burst of energy.
To repeat that, you would have to raise the skyscraper again. Which, guess what, takes at least as much energy as it released when you dropped it.Raising it back up to the same height to drop it again will take at least as much energy as you got from dropping it.
Also, building that skyscraper took a lot more energy than you can get back by dropping it.
The question is basically the same as asking why we don't ride bikes always downhill. Well, when we're up the hill we do. It's the getting them up the hill part that's the trick.
A: By convention, potential energy (which can be mechanical, gravitational, chemical, electromagnetic or nuclear) refers to energy stored in a field (electromagnetic field, gravitational field, gluon field etc.). This energy must be converted into kinetic energy in order to be "harnessed" or do work. For example, you can convert potential energy into:


*

*kinetic energy of an arrow, a pendulum or a pipe full of water

*heat energy (which at a molecular scale is just kinetic energy again)

*an electric current (moving electrons, so kinetic energy again)

*energetic neutrons and other products of fission or fusion (kinetic energy again)


So you can harness potential energy, but only indirectly.
A: The other answers give perfectly valid examples of e.g. a "grandfather's" clock, where you have to put in external energy from time to time - here, to lift the weight back up against gravity.
However, I feel that there is an additional aspect to the original question: The asker probably wants to stay completely passive and just harness the potential energy of the building, sitting on earth's surface, somehow.
To this, I say:
TL;DR: Theoretically possible, but practically meaningless (for now).
First of all, let's look at the basic formulae:
As the OP stated, a building exerts pressure on the ground. Pressure equals Force divided by Area:
$$P=\frac{F}{A}$$
So, to get the force a building exerts on the ground, we have to factor in Area.
No problem, the relevant area, the 'footprint' of the building certainly is known.
Now, what we want to get out of this is Energy. And Energy is Force times distance:
$$E=F\cdot s$$
 And that's distance downwards (hereafter denoted by $h$), because that's the direction the force works and therefore that's where Energy is to be gained. The force is, of course, gravitational force, so we finally get:
$$E=F \cdot h = m \cdot g \cdot h$$
Now, we've got everything we need. Drop a skyscraper (let's say $m = 200 000 t = 200 000 000 kg$) down, say, $10m$, in a controlled fashion(+) and you're looking at
$$E = m \cdot g \cdot h = 200 000 000kg \cdot 9.81\frac{m}{s^2} \cdot 10m = 19 620 000 000J = 19.62GJ$$
that you could harvest.
However, as you might have suspected by now, this is a one-time-only-ever-fullstop-period-type of deal. You can get that energy exactly once (++) and then, never again. You have to put up at least the same $19,62GJ$ to lift the skyscraper back up and repeat the exercise, gaining nothing in the process.
And factor in the energy expenditure from digging the $10m$ hole in the first place, putting up some mechanism to drop the skyscraper in a controlled manner and harvest the energy, you're certainly looking at expenditures that greatly surpass anything that is to be gained.
So... it's not really worth trying.
Addendum:
'Futurologists' propose that a similar technique might actually become feasible in the far future. It has been suggested that a sufficiently far advanced civilization could use black holes to collect energy. The principle is very much the same:
You feed matter into a small- to medium-sized black hole - which is equivalent to "dropping stuff down". But other issues are going to be easier, relatively speaking, with this concept:


*

*You don't have to worry about any structural integrity of anything.

*Preparing things should be "easier"; no hole digging required.

*You can, mostly, just forget about the matter that has crossed the event horizon. You just have to make sure that the gravitational influence of the growing black hole doesn't get out of hand.

*The collection of energy is very much easier (again, relatively speaking): Matter getting pulled towards a black hole gets really, really hot due to tidal and friction effects; up to a point where a significant fraction of the mass-energy gets converted into radiation. Then, you just need to collect that radiation via solar panels or equivalent, probably arranged like a Dyson swarm.


There you go. Using gravitational potential is, as of now, only a theoretical issue, but might become feasible on a stellar scale in a few millions of years(+++).
(+) just throwing it down would probably cause it to collapse and release additional energy originally expended while building the skyscraper, but all that energy would be pretty much non-harvestable...
(++) at least within our earth's lifetime
(+++) man-made satellites do a similar thing right now with their gravity assist maneuvers, but that's probably not the scope of this question anymore.
A: Simply No. If you could generate energy simply from the potential energy of the building, induced by gravity, without somehow permanently decreasing that energy, you would build some sort of perpetuum mobile.
If you would gain usable energy (like an electrical current) out of the potential energy of the building, without reducing the mass of the building and without altering the gravitational field, you would have created energy out of nothing, but energy is conserved.
A: An example of harnessing gravitational potential energy is a hydroelectric power plant which converts the potential energy of water falls, dams and the like into electrical energy. 
As far as harnessing the potential energy of a building sitting on the ground, I suppose if you caused the building to topple you could harness the energy of the falling portions of the building. Obviously ridiculous.
All practical examples of harnessing potential energy involve its conversion to kinetic energy.Hope this helps. 
A: Yes, you can convert the potential energy of the skyscraper into useful work. But, to extract useful work from the potential energy, must reduce the potential energy, that is: you must reduce the height of the skyscraper. You must tear the skyscraper down to get its energy.
You should note that skyscrapers aren't free and that someone used a crane powered by electricity or diesel to lift the parts of the skyscraper to their current positions. You are guaranteed to get less energy out of this process than was put in to build the skyscraper. You will waste a lot of energy in the process of converting energy from diesel or the electric grid into the potential energy of the skyscraper and then back into electricity. This would be a terribly inefficient way to store energy.
However, as noted by another answer, this is essentially what we do with hydroelectric dams. We move water from a high altitude to a lower altitude and extract useful work that is converted into electrical energy. This energy is free in the sense that the sun evaporated water somewhere and it rained down on the high altitude reservoir. So hydroelectric power is, at its core, solar power, because the sun effectively pumps the water uphill and we extract energy as it moves downhill.
Using actual electrically powered pumps, you can pump water uphill to store energy. You can use the energy later by allowing it to flow downhill.
A: As an alternative, you might be able to harness the energy in the thermal expansion and contraction of a structure over the course of a day.  Piezoelectric joints perhaps?
A: There is a company in Switzerland that is developing an Energy Vault, which is a building sized stack of heavy blocks with cranes extending out from a central pillar.
Each of these blocks can be lowered to the ground and the crane generates energy doing so.
Re-stacking the block requires using energy to take it back up.  There are, of course, losses when comparing the energy it takes to raise the block compared to the energy generated when lowering it.
This is meant to be used as an energy storage device, to give a method of storing excess "green" energy so that overproduction is not wasted.  Again, energy is wasted in this process, but that is true of any energy storage/conversion system.
Note that the stack of stones, when unmoving, generates zero energy.  It is only when the stones are actually being lowered that energy generation is possible.  The Potential Energy of an object is an ideal calculation of the Kinetic Energy that could be generated if the object was lowered a specific distance.  If the object is unmoving, no energy is generated.
A: In classical mechanics, absolute values of potential energy are meaningless. In your case of a skyscraper just sitting there, we could say it has a large positive amount of potential energy, no potential energy, or even negative potential energy. It doesn't matter at all. What is important is a change in potential energy.

is it possible to convert the potential energy of a building into a kinetic?

Based on what is said above, you would need to decrease the potential energy of the building and find a way to harness that change in potential energy. The issue is that for gravity, the potential energy just depends on the distance from the Earth, so this would mean that you would have to move the building (or at least parts of the building) closer to the Earth. The utility of buildings is typically that they remain stationary so people can use them consistently and for a long time, so I don't see this being feasible.
To see how gravitational potential energy can be converted to other types of energy in other systems, see some of the other posted answers.
A: We already do. Take a look at any hydropower plant in the world. The problem with trying to harness the potential energy of buildings is they would have to fall down. Being inside a building that's doing this would be disconcerting at best. 
A: As WaterMolecule mentioned, the key limitation is that harnessing energy, by definition, transfers the energy away from the source, eventually depleting it. The source, in this case, is the building's potential energy.
As user58973 elaborated, this depletion would look like the building sinking into the ground, and that's irreversible without giving back all the energy plus all transfer losses.
(Others mentioned tearing down the building. That's not necessary. Sinking it into the ground is enough.)
With this caveat, I argue that it's possible to harvest this energy, and then replenish it, to get the equivalent of a battery.
Very impractical and wildly uneconomical. But possible.
Picture a skyscraper built entirely on a sufficiently strong elevator. When the electric grid is down, the building starts to slowly sink into the ground. The elevator transmits that movement through a gearbox to run an electric generator, which powers the building.
When the grid comes back online, an electric motor slowly rises the building back, replenishing the potential energy. (it's possibly the same generator running as a motor, but, given the energies involved, you might want to go for specialized components.)
Talk to your utility before building. They may not be too happy about the double consumption after an outage, which could cause an overload and a new outage. But a single building in a large city should be fine. Then again, once they've let you do it, they'll have to let everyone do it.
If you can't get the relevant permits, can't spare the extra cash, or if you're boring, you can go for diesel generators like everyone else.
A: The only way to "harvest" potential energy from objects on earth is if a mass decreases in height from the center of the earth.
So if your building is stationary, you can't use it to generate power, but there are many things inside a building...
Some ideas I have for harvesting energy from the potential energy of the building (I'm not sure any of those are actually used):


*

*One-way elevators: make people walk up the stairs and use them as weight in the elevators when going down to generate electricity (elevator goes up empty - using some power then goes down loaded generating more power). Can work, but the people living/working in there will dislike you for implementing the system.

*Harvest rainwater on the roof, and make it go through turbines on its way down to the sewers.
A: Buildings often have water tanks on their roof, not least,  for firefighting. 
Thus, useful work can be obtained via hydroelectric pumped storage. Let the water fall, extract energy, re-pump the water at times when energy is cheap, or when (say) your buildingtop solar gets sun or windmill gets a gust.  
That may not be what you're looking for, but that's what works.  
A: Unfortunately, no. However large the building is, as its mass increases, the earth is just going to push back up harder to match the force the building pushes down (according to Newton's 3rd Law: every action has an equal and opposite reaction).
If we dropped the building from a great height it would certainly release a lot of energy - but you wouldn't want to work there.
If we rigged the building to a giant pendulum and released it, it would swing back and forth for a while but eventually slow down - just like any other pendulum. Note: a perfect pendulum would swing forever, but in the real world air resistance and friction are factors that would slow the pendulum down to an eventual stop. We can't get any more energy out of this whole system then what was put into it. That's due to Conservation of Energy, where the total energy of the system equals the kinetic energy plus potential energy of the system. So, when the building is at its highest point, it has the most potential energy but zero kinetic. When it swings past the lowest point it is going the fastest (highest kinetic), but zero potential energy, and then it swings up to a position of zero kinetic energy but highest potential energy. Even if you captured the kinetic energy at the bottom, you would be taking energy out. And if you take energy out, then the pendulum building won't swing as high, and the total energy of the system is less, and we're worse off then when we started.
A: You could harness energy from people using the stairs, for example.
If for instance you build your staircase with 1-cm coils at the base of each run, and the possibility for the stairs module to vibrate and slightly move along vertical guiderails, then you can feed a set of generators. The more people use the stairs, the more power you harvest.
Also, in tall buildings, water from gutters as well as sinks and toilets can be used to power small generators, as the water's running down vertical pipes with a sufficient amount of energy.
A: There is the concept of a "Gravity Storage" shown in this picture:
The potential energy of a huge rock cylinder is used to store electric energy. Whenever there is an excess of electric energy (e.g. solar energy when the sun is shining), the rock is lifted out of the ground using that energy. When there is a need of additional energy, the rock sinks back into the ground and the potential energy is converted to electric energy.
Theoretically the same would be possible using a building instead of a rock.
However, you have if the building is still in use and people enter or leave the building, you will not be able to lift or to lower the building for more than one or two floors (while you can lift the rock much more). Moreover, the rock is much heavier than the building.
This means that you cannot store too much energy using a building this way.
