This is an interesting question. I am sure that the capitol will not ask many types of questions, since they are an evil regime. Anything that would show this they will avoid. Here are some questions that they would most likely avoid.
First, they would probably avoid any question that deals with morality such as: "Is it simply wrong to allow children to kill other children?" Any question that surrounds this question will not be asked. In fact, anything that touches the question of the sanctity of life would be avoided.
Second, they would also probably avoid any question about how the contestants really felt. They would also avoid anything that allowed the people from the districts to speak their minds.
Third, they would also probably also avoid asking any question that surrounds the financial disparity between the districts and the central government.
They would avoid questions of morality and how they truly felt because if a question of morality arises, no doubt the Capitol is doing an inhumane act, but then the public would also be aware of this and revolt. If they were free to talk about their true feelings most of the contestants aren't pleased to take part in this because it is a question of life and death.