Warriors clinched a top-four spot but the five-try victory came at a cost with Weir carried off on a stretcher with a leg injury.
The club have revealed the Scotland international had been kept in hospital overnight for the injury to be examined with a further update expected early next week.
But coach Gregor Townsend knows the problem is serious enough to deny the 21-year-old the chance to be involved in the season finale.
Townsend told BBC Scotland: "He'll miss the rest of the season and probably be out for the start of next season. It's a real shame for him.
"To miss out on the last couple of games will be disappointing but there will be other players there to take their opportunity and play for Duncan."
Glasgow are likely to need a win away at Connacht on May 3 to have any chance of securing a top-two spot and thus a home semi-final.
But Townsend will be confident going into the play-offs no matter what happens beforehand. "Any of the teams in the last four have a chance of winning the title," he said.
"I would think we will have an away semi-final but whoever we get, we'll be playing tough opponents.
"If it's going to be Leinster or Ulster then we know what we have to do to win those games."