How does Zeffirelli deal with the problem that led to the heated debate during the following centuries in his film on Hamlet?
There have been, over the centuries, many heated debates about the play Hamlet. You might be referring the the question of Hamlet's Oedipal Complex and how Zeffirelli plays up the possibility of sexual attraction between Gertrude and Hamlet, but I will discuss the controversy surrounding the text of Hamlet and how Zeffirelli addresses that.
Much of the controversy stems from the fact that there are so many "versions" of the script. If one staged (or filmed) all the material that exists of Hamlet, it woud take 4 to 6 hours to perform. There are scenes (depending on Quarto or Folio consulted) in different order, additional text, lines attributed to different characters -- all of which have led to confusion and questions about the "real" script of the play.
As Zeffirelli did with his filmed productions of some Shakespeare's other plays (Romeo and Juliet being the first and arguably the most significant), he dramatically edited the text. For his film version of Hamlet, he arranged scenes and soliloquies in the order in which they served the story he wanted to tell, not based upon the most common order of presentation. The whole film runs a breezy two hours fifteen minutes, and is quite easy to watch and understand.
This approach to adapting Shakespeare for film with a strong emphasis on reduding the running of time of each scene, so that the story moves in a cinematic fashion rather than a theatrical one, was an innovation that Zeffirelli can be attributed with initiating. For my money, though his setting and costumes choices were not "modern," he was certainly the ground-breaking filmmaker who paved the way for films like Romeo + Juliet and the version of Hamlet starring Ethan Hawke -- films that keep the text 100% Shakespeare, but shape the story to suit a modern film-goer's sensibilities.