I put in de facto time limits all the time. The world keeps on ticking along whether the PCs do anything about it or not. I also use alternate rest rules, so it's not a blitz, more like "if we take a week off to get a long rest things will happen". So if you're in hostile territory, you can try to rest. It may work, or you might wake up to find yourself surrounded by a patrol. Decide not to track down that kidnapper so you can get a long rest? Oops, the mayor's son was sent home in several boxes.
One of the reasons I enjoy playing D&D instead of a video game is because video games are almost always static until activated. The real world doesn't work like that, I don't want my game to be like that either. Kill the patrol? When the patrol doesn't report in, things are going to happen. The PCs may not know what, exactly but it's probably not going to go well for them.
I don't normally do super tight deadlines, we're normally talking about a matter of days not hours. Once in a while it's also fun to have a sprint to the finish.