I think you play your guys. If they're nursing injuries, then obviously not those guys. But if they're starters or valuable contributors, they should play at least a half and you should keep trying to win the game.
What happened to Welker is obviously exhibit A in what could go wrong. It's also the worst-case scenario. But I think the benefits outweigh those risks. It's hard to quantify this type of thing since each year changes, but if we look at one team that regularly rests their starters to the point of giving up on a perfect season, we can measure this a bit more consistently.
For 5 of the past 6 seasons, the Colts have started Manning for a series or two and then pulled everyone in the last week of the season. The only year they didn't was 2006 when they were trying to win the game for home-field in the event they met up with the Patriots (which they did). No surprise, that was the only year they won the Super Bowl.
In 3 of the 5 seasons they rested the last week, they lost their first play-off game, despite hosting 2 of them. In one other, they won 1 game before losing to the Patriots. The other game was last season when they went all the way to the Super Bowl, but that was also a bit unusual in that they rested almost a full month by waving the white flag two weeks early instead of one.
On the other hand, Brady hasn't rested much in that last week during his career. The only time he really got a chance to rest was 2005, when he threw just 8 passes against Miami before being removed from the game. We beat Jacksonville in the wild card before losing to Denver.
In his 8 seasons, he played most of the game to try and win 7 times, and his record in those games is 6-1. Last year was his first loss in the last week of the season when he played the majority of the game. Coincidentally, it was also the first time in his career he was one-and-done. That had more to do with Welker's loss though.
And as much as I hate to bring up the 2007 Giants, give credit to them for playing hard in week 17 when that game meant nothing to them, and they lost a few guys to injury. But it gave that team confidence going into the post-season. People forget how streaky that team was. They lost their first 2 games, won 6 in a row, then went 4-4 down the stretch, alternating between wins and losses (L,W,L,W,W,L,W,L). They were a team filled with questions, but that last week seemed to bring them together for their play-off run.
Losing Welker really sucked. But a lot of signs point to playing to win that last week.