There are no requirements as such. Nobody made a list of languages and said :"OK, if Cardinal X hasn't that one checked, he's out of the question".Up to the 20-th century, Popes didn't travel much, and media was not that powerful., so Latin and Italian were pretty much sufficient.After Benedict XV (died in 1915, I think) Popes started to come from the background of Vatican diplomacy (i.e. they were former Papal Nuncios) so they at least knew French(the language of diplomacy at the time), plus possibly the languages of countries which they had spent more time in (like German for Pius XII, Turkish and Bulgarian for John XXIII). But then after WW II it was media age already. Paul VI notably learned English during his pontificate (he broke the short-lived tradition and hadn't been a nuncio, just a diocesan bishop). Little did the Cardinals know when they elected JP II that he actually could speak 14 languages (but each one of them was content to hear him speak in his language, LOL).
OK, enough with history:here's the present facts:The Curia (the Vatican offices) works with Latin and 7 modern languages:Italian, French, English, German,Spanish, Portuguese and Polish(which represents all the Eastern European languages). I'd say, if someone knows all these it's a good qualification :)
Benedict XVI knows all these, plus notably Czech (as an Archbishop of Munich, his diocese bordered on Czechoslovakia and he was involved in actions to help the Church there). I think the one he speaks best beside German is French(absolutely no accent).
For the other languages (as recentlyfor Turkish) he has his speeches translated and written phonetically)