Does the body burn calories repairing muscle?

Does the body burn more calories while idle if repairing fatigued muscle than if there was no muscle to repair?
Any benefit to taking 1-2 minutes before going to sleep at night running a muscle to fatigue. Say doing push-ups until you can’t. Then letting the body burn calories while you sleep repairing the muscle?
Or is this premise completely flawed?

Well, your body IS, technically, doing more work repairing the muscle then if it wasn’t, but I can’t imagine it would be any significant amount. Not to mention that often times doing something till muscle fatigue isn’t always because you damaged the muscle and it needs repair, but you just used up all the muscle’s energy reserves. That’s why even after lifting weights till fatigue, you can just wait five or ten minutes and lift again.

Whatever calories you burn repairing muiscle is a drop in the ocean compared to the ones you’ll burn just doing the push-ups, I’m sure.

The more general principal being alluded to is excess post-exercise consumption (EPOC). There are calories burned recovering from exercise, especially in the first several hours after exercise, and more burned after more intense exercise even if that exercise was of shorter duration and less total calories burned during the activity than more moderate intensity activity. Some of EPOC is indeed cellular repair. Some is replenishing energy stores. Some is building new tissue (anabolism). EPOC does use up fat and is why the garbage about low intensity exercise burning fat more (because it does so during the activity more than high intensity exercise) is just that: garbage. Over the day you will burn off more fat with having done more intense exercise.

Some do believe that EPOC should mean that having two more intense periods of exercise of shorter duration would have more fat burning effect than one bout twice as long of the same intensity. That’s sort of the concept the op is proposing as well: do a short bout to rev up and recover from for a few hours. I don’t know if 1-2 minutes is enough, but 10? Enough to fit in a set of Tabata intervals (maybe doing burpees) with a brief warm-up and cool-down? Yeah, that would have real benefit.

I don’t know of any reason why the exercise would have more benefit at bedtime than at any other time. And exercise too close to bedtime can interfere with sleep.

On a similar note, I’m currently doing an exercise regime that’s pretty tough for me. Sometimes I have to rest in the middle of it. I’ve always wondered if I would burn more calories if I toughed it out for 5 mins, or rest after 2.5 mins and do the other 2.5 mins after a minute of rest.

This doesn’t exactly address the OP’s question about repair after exercise, but there are good numbers on how the body can expend huge amounts of energy repairing significant damage. In the aftermath of surgery, broken bones or the like, even patients with bed rest may be burning 20-30% more calories than their usual basal metabolism. Burn victims can double the calories.

Of course, the damage being repaired in those examples is far more significant than just a tired muscle, but maybe it will help provide a context.

This article gives many details on “Exercise After-burn” (EPOC) and pertinent to your question states:

So in principle splitting it up is a better choice, and even more so if you keep up a higher intensity for the splitting.

The whole article is very good for anyone interested in this kind of stuff.