The New Yorker has a good piece on quotations, and the first paragraph focuses on spurious quotes--that is, Patrick Henry really didn't say "Give me liberty or give me death."
I can think of a couple of semi-spurious quotes: one is that nobody coined "He's a son-of-a-bitch but he's our son-of-a-bitch," until the 1960s; somebody, maybe Lyndon Johnson, used it to refer to Ngo Dinh Diem, but he probably didn't use it first. Latin American folklore says that quote was used in the early 20th century in reference to the first Somoza, but I've never seen any proof.
Another is "America for the Americans," which does not appear in the Monroe Doctrine, and was actually used as a Nativist anti-immigrant slogan in the 1840s or so. It has nothing to do with Latin America, either; the immigrants in question were Irish and Germans.
One from Spain is Unamuno's "You will win, but you will not convince" speech in Salamanca in 1936; he was extemporizing without notes and nobody was ready for him to say anything of note, so reports of the speech were all written down later. There is no recording. So we really don't know exactly what he said, though whatever Unamuno did say caused quite a tumult in the auditorium. Unamuno died just weeks later, so he wasn't there to ask later. By the way, everybody seems to think that in the Civil War Unamuno was pro-Republic; he was actually pro-National, though with reservations.
Another one from the Civil War is that no one knows exactly what was said during the famous phone call made by Colonel Moscardó's son, under duress as he was a Republican prisoner, to the colonel, who was leading the National resistance at the Alcazar in Toledo. It's generally agreed that the phone call was made, that the son told Moscardó that he was going to be shot, and Moscardó told the son to die bravely. But the exact words spoken are unknown.