MTGNexus

What happens if I exert a creature and then Donate it to an opponent? What happens if the opponent donates it back?

Exert cares about "your" untap step where "you" are the person Exerting it. So, if you Exert it and Donate it, it will untap during their untap step as normal since that isn't your untap step.

However, if they Donate it back, assuming your next untap step has not yet occurred, it will not untap per the Exert ability. If they Donate it back after your next untap step had occurred, the duration has expired and it will untap as normal. So this one depends a little on the timing but it is both possible for it to remain tapped or untap depending on how quickly you get it back.