In the movie, Johnny Galecki's character Borel, gets 10 years from Will Salas and drinks himself to death. His wife said that he had 9 years left when he died:
Borel, open up. it's Will.
You gave him a decade.
He drank himself to death
with nine years on his clock.
Greta, I'm so sorry.
This doesn't explicitly say whether the time continued running or not, but there was no mention of some scramble to steal the remaining time (not that that was the point of this scene).
There is also the scene where Will Salas and Silvia Weis crash the car. There is this bit of dialogue:
The girl's still breathing.
She's got a decade.
It seems that there is no point of trying to take time from a dead body. (However, even if she were dead, the comment "She's got a decade." would not seem out of place.)
Perhaps it can be inferred as the remaining time is irretrievable, and thus if a person dies before their time is up, the remaining time is lost. Since small details like this were not the focus of the movie, it is unlikely that there is a movie-canon answer.