"František's aunt likes my hands."
Translation:Františkově tetě se líbí moje ruce.
What do you mean by "dat + acc?" Can líbit have an additional direct object? Or can the accusative be used instead of the dative? I have never seen either and would appreciate an example.
My fault. It should be líbit se + dat + nom. That's the pattern. You use dative to say who's the person (in this case Františkova teta) who likes something (moje ruce). That something is in Nom.
If your question is "why not genitive (Františka)?": "Františkově" is a possessive adjective, something that is used only for proper names (a bit like Cesarean for Caesar or Neronian for Nero – Franciscean?). Its nominative singular is "Františkův"; it is in dative here to agree with "tetě."
The grammar of líbit se, with dative, is explained excellently by Endless Sleeper's comment.