If I originally lived in Italy and then moved to the United States and currently live in the United States. Then somebody asked me where I am from. I say:
I come/came from Italy.
Should it be present tense or past tense?
"I come from Italy" is the better semantic choice. "I came from Italy" refers to a more specific time frame: "I came from Italy last night." However, the more natural one would be "I'm Italian but currently live in the States."
Also, the past tense usually refers to an action that is 100% complete and done with; however, "I come from Italy" shows that the relation is still an existing one.
Normally, it would be "I come from Italy", "I am from Italy" or "I'm Italian".
It's technically correct for you to say "I came from Italy", but that would sound like you're recounting the the journey. But if you are actually talking about your experience of arriving in America, then that would be correct:
I came from Italy in 1983, and found I liked it more than I expected.