Why is it harder to open the cap on a bottle than to close it tightly? In my experience it's harder to turn the tap off a bottle than to put it tightly on. Why?
 A: Because there is two types of friction, static and dynamic friction.
Static friction is always greater than dynamic friction. Static friction opposes motion of the object while it is not moving. Dynamic friction is what opposes motion of the object while it is moving.
In a closed lid, turning the bottle cap requires more force because the static friction requires a large force to overcome that friction.
When closing a bottle with a lid, you are moving the cap from no-friction (no contact with the bottle) to dynamic friction. This is why it is easier to close than to open a cap on a bottle.
A: Having tested it, using maximum force screwing it on, I always managed to open it again. The test was repeated with my other hand btw, to make sure it wasn't due to a difference in strength between the muscle groups involved in both actions. Still not perfect, because grip is the limiting factor (strain-sensitive receptors in the skin detect the amount of friction, and we automatically apply enough grip force, with a safety margin, to prevent slipping), and we can't be sure the caps aren't designed to provide more grip in one direction than the other (seems a good idea in fact..). 
To get an objective measure, I nailed the cap to one end of a piece of wood (about 3*4*80 cm, the cap on the 4*80 side), and, holding the bottle horizontal, turned it until the beam lifted from the table. Then I turned the assembly around, and turned the bottle in the other direction. The beam would lift from the table for 1 to 2 seconds before the cap came loose. Repeated tests, and tests starting with the beam hanging down, slowly rotating until horizontal, all gave the same result: the torque that closed the bottle would also open it.   
So what is going on? I've had bottles that were almost impossible to open as well, and my ice tea doesn't contain CO2, so gas pressure can't be the explanation in that case. The difference between static and dynamic friction mentioned by Kenny Guy would not explain why, at least in my experience, the cap in such cases slowly begins to turn when enough torque is applied, but initially the torque required to keep it moving doesn't decrease much. If static friction was the cause, you'd expect the movement to be more sudden since the friction would drop abruptly. 
I'm thinking it could be due to non-uniform stress relaxation: Stress relaxation is like extending a spring, keeping it at that length, and the spring losing its tension over time. When two springs are connected in series, and only one of them undergoes stress relaxation, then that one will eventually be stretched over the total displacement. 
Real stress relaxation in polymers isn't 100%, but it would have an effect no matter the amount. 

If stress relaxation mainly affects the thread of the cap, then the segments facing the notches in the bottle thread, which wouldn't be affected since they aren't under load, would end up deeper inside the notches (since the strain of the cap is transferred to the thread, the cap "shrinks", pulling the unaffected part of the thread upward). It would explain why the torque only decreases after turning a certain amount: the stuck thread has to come out of the notch. Just an idea, but who knows.  
