In speed chess, if my opponent's clock time per move runs out, do I get to take a piece?

No, you do not get to take a piece. If you are playing with an increment, for example 10 | 5, a ten minute game with an additional 5 seconds per move which is extra time for the players to play the game. The extra time per move is so that if the players play a 100 move game, they will have more time, but still play a fast paced game. When the extra time runs out, their base time starts to run down. If your opponent runs out of time, and their flag falls on the clock, you may claim a win on time, you do not have to claim a win on time.