There's no reason for Italy and Iran/Iraq to be pronounced similarly. (Cf Indiana, Illinois, Iowa, Idaho?)
But FWIW, the EYE-rack thing is because GWB (most prominently, but others before and after) intentionally mispronounced the name of the country, in a "real american" kind of way, and also to annoy SAD-dumb Hussein as a kind of "we're stupid but we're going to kill you anyway" kind of psyop. Or maybe just "we disrespect you in advance of killing you"?
Americans of other political persuasions usually pronounce the names correctly.
I've lived in over a dozen states and I've never heard either called anything other than EYE-(ran/raq) in conversation.
The extremely, I mean extremely rare occasion when someone pronounces it differently on TV, it's almost like they get side-eyed by other people as trying to "talk fancy".