1982:
The flashbacks from "Batman: Mask of the Phantasm" are set in 1982. We know this because the events portrayed in these flashbacks are obviously set between the years Bruce trained with Yoru Sensei (1980-1981) and when Bruce (who was already Batman by that time) brought Robin into his care (1983).
The story that unfolds in the present section of the movie is set in 1993. We know this because the Bat Signal is used during the story, and a line of dialogue reveals that it's been used several times before.
The Bat Signal isn't introduced until an episode called "The Cape and Cowl Conspiracy" (4 episodes before this film), which is set in 1993. However, during an early scene in the movie set in the present, Andrea Beaumont says:
This makes it seem like the flashbacks are set in 1983, but that's not the case. This isn't a continuity flaw. Andrea is reminiscing about the last time she was in Gotham; she's referencing an event rather than a particular year. What I mean is, when she says 10 years, what she really means is less than 11 years have passed since she was last in Gotham. Which means that even though this scene is set in early 1993, her line can actually be referencing events from late 1982.
In the the flashbacks to 1982, Bruce encounters his inspiration for the future design of the Batmobile:
He's also inspired by a sudden fear of bats, that reveal the future Batcave:
