What does "I must have you dance" mean?

What does "I must have you dance" mean? What kind of a sentence is it?

The only way I can make sense of that sentence is someone expressing his or her strong desire to induce the dancer to dance for him or her. Although not all may agree, I think of this sort of construction as a part of high-society small talk.

[polite conversation]
1: I quite enjoy the arts, don't you?
2: Yes, I certainly do. As a matter of fact, I'm a dancer.
1: Wonderful! I own a dance hall, I must have you dance sometime.

This is a pseudo-passive construction, I’ll have you know.

The question here is on the polysemy of the verb have.
Followed by object pronoun + past participle or bare infinitive have can mean make or oblige s.o. to.
It implies that the grammatical object (you in your example) is not entirely willing of the action expressed by the grammatical subject (I in your example).

"I'll have you do your homework, whether you like it or not." (says mum to son)

"I'll have you executed on the spot."

(King of Hearts to Mad Hatter)