Warburton was hurt in the dying seconds of a game that saw Wales run in seven tries - six of them after Scotland full-back Stuart Hogg was sent off.
Wales head coach Warren Gatland confirmed the extent of Warburton's injury, and said he will undergo a scan on Sunday.
Gatland said: "I don't think he (Warburton) is very good.
"It looks like he has dislocated his shoulder. It was put back in after the game.
"He will have a a scan tomorrow, and the likelihood is he will need an operation on that and be ruled out of the rest of the season and probably the tour as well."
Last year's British and Irish Lions captain Warburton left the Millennium Stadium with his right arm in a sling, and it is the second major injury blow for Gatland inside a week.
Warburton's fellow Lion, full-back Leigh Halfpenny, dislocated his shoulder completing a try-saving tackle on England centre Luther Burrell at Twickenham last Sunday.
Halfpenny underwent surgery two days ago and is expected to be sidelined for up to 16 weeks, meaning he will miss June appointments with South Africa in Durban and Nelspruit.
Wales' Six Nations title hopes disappeared through away defeats against Ireland and England, but they ended the campaign with a third successive home win.
It was their record Six Nations victory, surpassing the 47-8 stroll past Italy in 2008, and also a record win against Scotland, doubling the previous-best winning margin at Murrayfield nine years ago.
But the game will largely be remembered for Hogg's 22nd-minute red card, which came after he hit Wales fly-half Dan Biggar late, smashing him in the face with his shoulder.
Referee Jerome Garces initially showed Hogg a yellow card, but he changed it to red after watching replays on the stadium's giant screens.
"You've got to feel sorry for Stuart because he is not that sort of player at all," Gatland added.
"It is a rush of blood, and he's made a mistake.
"From our point of view, we are not going to dwell on that. I don't think you will see that from him again.
"He has made a mistake. It's a collision sport. From our point of view there are no hard feelings.
"You have got to be ruthless, and we were ruthless from that point on. Even before that, I thought at 13-3 up we were reasonably comfortable with the way the game was going.
"We kept the ball, and kept it for long phases, and I didn't feel Scotland could handle our physicality in defence or in attack.
"We know in this tournament we've been a little bit inconsistent. I thought the players responded particularly well today, and it was a good display."
Reflecting on the Six Nations campaign as a whole, Gatland said: "We've done reasonably well in this competition. We've set massive expectations upon ourselves.
"The media put a lot of pressure on us, but in the last three years we've won 12 out of 15 (Six Nations) games. I don't think that is a bad record in this competition for a small country like Wales with such a small playing base as well.
"We are pretty happy with where we are at the moment, but as I said, there is no-one harder than we are on ourselves in terms of being critical - be it coaches or players - because we want to keep working and improving as a side.
"We have a really tough tour of South Africa in the summer.
"We will take a slightly bigger squad, and we are looking potentially at a midweek game against Southern Kings in Port Elizabeth, so everyone is a part of the tour. We know how tough a place it (South Africa) is."
- Sports & Recreation
- Warren Gatland
- Stuart Hogg