Does the relative speed of time mean there is less energy where time is slower? Time runs relatively slower near a planet than in outer space.  Does this mean that there is less energy near the planet?  Is there a relationship between energy and the speed of time?
If so, this would seem at odds with the slowing of time as an object speeds up.  Wouldn't an object at relativistic speeds have more energy than an object at rest?
 A: There is no contradiction:
In a weak gravitational field, with the approximate expression of a constant field, we have $$\tau=\frac{x^{0}}{c}\left(1+\frac{\varphi}{c^{2}}\right)$$ with $\varphi<0$.
In a weak field, we have the approximate formula $$\omega=\omega_{0}\left(1-\frac{\varphi}{c^{2}}\right)\;\;\;\;\;\;\;\;\;\;\;\;(1)$$
We see that the frequency of light increases as the absolute value of the potential of the gravitational field increases, i.e. as we approach the bodies producing the field.
It can be demonstrated that the energy of a photon increases in a gravitational field without using general relativity (Einstein's argument in 1911), a photon approaching a source of the gravitational field has an energy  $$E=E_{0}+mgh=E_{0}+\frac{E_{0}}{c^{2}}gh=E_{0}\left(1+\frac{gh}{c^{2}}\right)\;\;\;\;\;\;\;(2)$$
(1): Volume II of L.Landau,E.Lifchitz , Field theory.
(2) :Gravitation,  Carle W.Misner, kip S.Thorne, john Archibald Wheeler.
A: Two positively charged plates are pushed closer together using a constant force F that moves the plates distance d. Energy used is:  E=F*d.
Then the system of two plates is accelerated to speed 0.87c at which speed the Lorentz factor, or time dilation factor, is 2. The repulsive force between the plates is now F/2, in the frame where the plates move, we can say that it's because of magnetic repulsion of the plates. Energy released when plates move to the original relative positions is (F/2)*d. 
That released energy can converted to a piece of matter that has energy E in the frame of the plates. 
In the frame where the plates move, the piece of matter has total energy 2E. 
Now, how should we divide that total energy 2E between kinetic energy and non-kinetic energy?
The released energy was 0.5 E, if we attribute that energy as the non-kinetic energy, then the kinetic energy must be 1.5 E. 
That kinetic energy is larger than the energy that is released when we stop the piece of matter. After the stopping what we have is: released energy E, and a piece of matter with energy E.   
A: 
Time runs relatively slower near a planet than in outer space. Does
  this mean that there is less energy near the planet?

There is less energy if you consider the minus sign of the gravitational potential energy, however, its magnitude (absolute value) is much.

Is there a relationship between energy and the speed of time?

Yes, $E=h\nu=h/T$.

If so, this would seem at odds with the slowing of time as an object
  speeds up. 

No, I do not think so. The time as the internal energy of the system slows down and changes into the external energy (kinetic energy) of the system. Moreover, as stated by stuffu, a part of this energy is converted to matter too.

Wouldn't an object at relativistic speeds have more energy than an
  object at rest?

Yes, it would. Therefore, I think you can draw the following conclusions:
1- The energy magnitude near a planet is great, and thus time (internal clocks attached to the planet) runs slower.
2- The energy magnitude of a uniformly moving object is also great, and time (internal clocks attached to the object) runs slower again.
For more clarification, assume that you have a tiny planet with a clock attached on its surface. The more you increase the mass of the planet by, say, adding some matter to it, the slower the clock runs. Here, adding matter means increasing the (potential) energy of the system. Consequently, we can infer that the more the velocity of an object increases, the slower the clocks run. Here, increasing the velocity means increasing the (kinetic) energy of the system.
A: I didn't get exactly what caused you to think that slower time corresponds to lesser energy, but the general fact is that more energy causes more gravity, which causes more 'slowing' of time. So if time flows slower near a particular place, it is because of more energy present there.
Normal time dilation, caused by high speed, is different than gravitational time dilation, which occurs due to effects of gravity. So there is very little connection between them, except for the fact that they arise from the same theory. But yes, in both case more energy corresponds to more 'slowing' of time, although this is the consequence of different mechanisms.
A: The comparison with objects at relativistic speeds has the problem of symmetry. For each one, the time is slower in the other frame.
On the other hand, an observer in a gravitational potential has its time slower than an observer far away, and it is confirmed by the other frame.
But if we compare with an uniform accelerated frame, we have the same kind of true time dilation than a gravitational potential.
We can think of a ship moving like an oscillator to an inertial observer. Maximum (and relativistic) speed, when they meet and no acceleration. Just after that, turn on the engine (keeping an acceleration $g$ to be comfortable) to reverse the velocity, until get $v=0$ relative to the inertial observer, and return. Just before meet again turn off the engines and repeat the process at the other direction.
Every meet time, both observers can agree that the accumulated time in the accelerated frame is smaller, as he was in a gravitational potential. And if we remember of the equivalent principle, he is indeed.
But it is only possible by burning fuel in the ship. So, we can say that the accelerated observer decreased the energy content of its ship (measured in its own frame), to be younger than the inertial observer.
Following that idea, in both ways (gravitational potential or accelerated frame), aging slower requires losing energy.
A: From Wikipedia:

If a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

In GR, the first part of this statement is true for sure. So the energy of a system is conserved (at whatever place) if we make a translation of the Lagrangian in time, no matter the pace of time actually is. So at every point in spacetime energy is conserved, regardless of the pace of time, which means there is not less or more energy contained in a physical process where time has a different pace.
