When you first start out doing a particular exercise, obviously it's harder than if you've been doing it for a while and your body is used to it. Does that mean it takes more energy, and that you therefore burn more calories doing an exercise when it's new to you, compared to when you're "in shape" for it?
No, you are burning fewer calories than when you are experienced--particularly if you are increasing load.
Based on my own experimentation with my Polar heart rate monitor I have found that the calories burned is a function of the amount of work being done. The more experienced you become, the more adapted your body becomes to the type of work you give it. Your range of motion improves, the amount of additional load (weight) you can handle also increases, and the amount of work you are doing increases. As you get stronger, you either increase your muscle mass or become more efficient using them. This in turn improves your metabolism, which will burn more calories even while sitting still.
The reason a new exercise is hard at first is because your muscles are not used to working that way yet. You can't use all the potential energy you have until you acquire both the skill (muscle coordination) and strength to do it properly.
No, experienced athletes burn the same number of calories as novices when performing the same action over the same time.
We will use physics to answer this question. In particular, the forumla for potential energy will be of use:
Let's take the simple example of lifting a 20 kg dumbbell from the ground to 1 meter high.
It takes 196 joules of energy to increase the potential energy of that dumbbell. Where does this energy come from? It must come from somewhere to satisfy the law of conservation of energy.
The energy comes from the body burning calories. It must burn 196 joules to comply with the law.
You burn 0.00467 kcal to move that dumbbell up. (Food items are always labeled plainly with calories, but technically they mean kilocalories). No where in these equations does experience or ease of effort factor in. Arnold Schwarzenegger in his prime will burn 0.00467 kcal and a scrawny kid will also burn 0.00467 kcal to move a 20 kilogram dumbbell up 1 meter. Yes, Arnold will have a easier time because trained muscles store more glycogen and stronger muscle fibers contract. But he cannot defy the laws of physics by burning less calories than the scrawny kid.
The above assumes that the experienced athlete uses the same technique as the novice. In the real world, this is not always the case. In nearly every sport except bodybuilding, athletes use special techniques to more efficiently use their energy. These techniques channel more energy towards there goals and lets less energy dissipate for other means. For example, I recall in the 2008 Olympics, the announcers said Usain Bolt took on average less steps than his competitors. This allowed him to more efficiently channel his energy towards running fast. His competitors were generating less speed for the same amount of calories burned.
Movement efficiency can be improved though training. How much of an effect this has depends on the form of exercises but generally isn't that big.
But the situation is a lot more complex. Let's say you go running. Over time you might loose some weight, which means you use less energy to run the same distance at the same pace. On the other hand as you get better at running you might run the same distance in shorter time, which uses more energy.