"Do you like dancing, too?"
Translation:Tanzt du auch gern?
1) While it is not idiomatic, it is not wrong per se. But:
2) It sounds very much like Magst du (auch) tanzen? which is polite way to ask for a dance.
3) If you want to use the verb mögen, you need to introduce an impersonal es:
Magst du es auch zu tanzen?
"auch" is usually written (or said) in front of the part which it refers to. In this case it emphasis the fact that one likes to dance. If I wanted to emphasis "du" I could say: "Tanzt auch du gerne?" But this is not the meaning of this duo sentence.