Yes you can say that for any exam, it doesn't have to be a written one. However your sentence should be "I have to take a dance exam tomorrow" or "I have a dance exam tomorrow". Another way to say this phrase would be "I have an exam". For example: "I have a dance exam tomorrow" "I have an English exam next week"

"I have to take a dance exam tomorrow" doesn't sound natural. Exams don't have to be written (they can be oral) but they test knowledge. Performances typically have "tryouts" or "auditions". To get a drivers license, you have to pass a written exam (or test) and also take a driving test (not exam). Now it's possible, for example in a school setting, that "exam" is used more broadly, but I'd be wary of using it that way.

Yes, you can take all kinds of exams, but note that you have a dance exam -- you need the article, at least in American English. I should also note that this sounds rather odd, as if you had a written or oral exam on dance theory, not a performance that was to be examined.

Sorry, but as a professional musician I had to correct the idea that a dance exam sounds odd. Exams in the performing arts are common. Auditions are usually to get a job, or a place on a course. Exams are to gain a qualification showing your attainment of a certain level of skill. Violin, drama, singing, ballet... you can 'have to take an exam' in any of them. Same with other practical skills - woodworking, upholstery, driving - practical exams are part of the qualification process. For anyone who works in dance, the phrase 'I have to take a dance exam tomorrow' sounds as natural as 'I have to go to the bank tomorrow'. More 'natural' English, though, is probably to omit 'to take': 'I have a dance exam tomorrow and I've lost my ballet shoes!'

Examination in creating