Tapping is the most important move in Brazilian Jiu-Jitsu. It makes it possible for you to keep training and competing for years to come, and it also opens up more learning and rolling opportunities.
However, are there any instances in which you should not tap out? John Danaher shares his opinion:

First off, in training you should be tapping very early. Because you’re not getting paid to fight in the gym, you’re getting paid to fight on stage.
So, be a professional in the gym; tap often, tap fast. That way, you’ll last a lot longer.

However, things change a bit when it comes to competition:

You’ll have to specify what is the specific situation.

Alright, so if you’re in the first round of the ADCC, your first match and you get caught – I would always expect my students to tap. Because, if you get your leg broken or your arm broken in the first round, you still have three more matches until you get to the final.

I also think that if you’re in a strangle hold, it doesn’t make a lot of sense in not tapping because you’re going to pass out. I don’t see any heroism in just letting yourself pass out.

Danaher emphasizes that competing in the ADCC finals is a whole other thing entirely:

Now, things change if you get to the final. If you’re in the final and you’re ahead in points… For most people, at that point they’re willing to let something break to win the gold medal. At that point, I leave it up to the student.

It’s a deeply personal decision. I would never say to a student: “I expect you to let your body break.”
I think that my students are more than mature enough to make up their own minds.