How long did it take Burt to learn/speak Italian?

Lancaster didn't learn to speak Italian for the film, he spoke his lines in English, as did some of the other actors with whom he performed. In fact, Delon and Cardinale spoke English in the scenes they filmed with Lancaster, French in their scenes together, and Italian in the rest. Not a little confusing, huh? Luckily, Cardinale was fluent in all three languages, though, since she grew up speaking French, had to be dubbed by another Italian actress in "The Leopard" and her other early films. (Her French-accented Italian was supposedly off-putting to Italian audiences' ears.) The French DVD of "The Leopard" has the original French soundtrack where both Cardinale and Delon can be heard speaking their own lines.

And don't forget that in the italian version Lancaster is dubbed with an heavy sicilian accent... and that isn't fully "italian", even regarding some syntax. :-)

It took him about four hours of heavy reading of "Italian for Dummies".