From the perspective of Person A does time speed up for Person B the faster Person B travels?

Answer 1

No; just the opposite. From A's perspective, time has slowed down for B, because of his speed. (This is only measurable for appreciable fractions of the speed of light, but has been measured with super-precise atomic clocks flown around the world in an airliner. )

Curiously, B also perceives that A's time is slower, because from his own perspective, he is standing still while A is moving. And relatively speaking, both perceptions are equally valid! Incorrect. Both perceptions are not equally valid because one frame is an inertial frame of reference while the other is non-inertial. Only the perspective of the inertial (rest) frame is valid.

Answer 2

From the perspective of Person A, time does speed up for Person B as they travel faster. This phenomenon is explained by the theory of relativity, where time dilation occurs as one approaches the speed of light. So, Person A would observe Person B's time passing slower than their own as Person B travels faster.