Potential problem would be when leaving Italy ...
My stepson is Italian and has a different surname from mine, and when I travel with him according to Italian law I need a "delega" from my Italian wife who is his mother, when passing passport control flying out of Italy. Now he's 11 and quite tall (155cm), this is rarely requested, but when he was 6, 7, 8 etc I always had to show the piece of paper. The official document is obtained from the Italian consulate but for a foreign national, a signed attestation "to whom it may concern" is enough.
The rules were introduced after some high-profile cases of people abducting children out of Italy to I think the Middle East.
In your case it would be enough to prove that you're her father, assuming you are (i.e. not her stepfather). I would certainly suggest taking her birth certificate in that case. Nine times out of ten it isn't needed but you never know.