The Yankees took a 7-0 lead in that game. It's not like they weren't trying. Sure Girardi pulled most of his starters, because why overwork your starters when you play in 2 days and you don't need the win?
Still, even with pulling his starters and nearly blowing the lead in the 8th, they had the Rays down to their final strike with no one on, down a run, and the pitcher wasn't throwing softballs up there.
Could they have thrown Rivera out there? Sure, but why?
They weren't trying to lose the game. The players in there at the end were clearly trying to win. They just didn't use all of their best because they were resting some of them for the playoffs.
If they get in a 17 inning game or something with their starters all running out there for 7 or 8 at bats, and the bulk of their main relievers getting the work, they're tired out for Friday's game that actually counts. You can't fault the Yankees for any of that. They didn't use all of their best, but they definitely tried to win the game and nearly did.
ETA: It's no different than what some coaches do during the SEC baseball tournament when they already have a national seed locked up. You'll see teams hold their ace for a day or two to keep him on a normal rotation rather than pitching their ace in the opener, because the game doesn't matter to them. If they're playing a bubble team, is it fair to other bubble teams that they're giving that team a good shot at a big win while facing a midweek guy? Maybe not, but you coach for your own team's situation. If you have everything you need clinched, you can rest players and set your rotation.