If a committee feels that a bill is not important, they push it aside and let it die. What is that called?

Veto

Explanation:

Pigeonholing. the procedure during which the chairperson's leadership and the committee can ignore a bill and simply let it die. Hearings. a session at which a committee listens to testimony from people interested in the bill. Pocket Veto.

If he breaks the law then he had to resign from his post and so he'll lose his/her powers.