Cornelius Fudge was last seen after resigning at Albus Dumbledore's funeral. What happens to him after that?
We don't know.
The last time Cornelius Fudge is seen in the Harry Potter books is at Dumbledore's funeral at the end of book 6. That's a whole book after his resignation as Minister for Magic (after seeing Voldemort at the end of book 5), but we hardly see him at all in the rest of book 6: I think the only sight of him there is in book 6 chapter 1, narrated from the viewpoint of the Muggle Prime Minister, when he's revealed to still have at least some role in Scrimgeour's Ministry, acting as a liaison between the new Minister for Magic and the Prime Minister.
(In the Harry Potter films, a Daily Prophet article suggests that Scrimgeour met personally with the Muggle Prime Minister, so it could be that Fudge eventually lost this liaison role. But a newspaper seen briefly in a film adaptation cannot be used as very conclusive evidence for what happened in the books.)
It's unknown what happens to Fudge when Voldemort takes over the Ministry in book 7. Scrimgeour dies, but other Ministry figures such as Umbridge don't, so we can't conclude anything about Fudge's fate. It seems unlikely that he'd become so eager a collaborator as Umbridge - his character as Minister was more incompetent and ambitious than wantonly evil - but he might have escaped, or been Imperiused, or been able to continue some harmless Ministry duties. (Did he continue liaising with the Muggle Prime Minister? As far as I know, it's never mentioned what happens to the Ministry of Magic's connection with him during its time under Voldemort's control. Given Voldemort's attitude towards Muggles, it's unlikely that the connection between his puppet Minister of Magic and the Muggle Prime Minister is maintained. Perhaps Fudge found himself without a job.)
The inspiration for Fudge's character, specifically his actions throughout book 5, was Neville Chamberlain, as the author said in a 2008 interview:
About the relationship between September-11 and Harry Potter, Rowling denied any influence and recognizes who was the true inspiration for the Minister Cornelius Fudge: "My model of the world after Voldemort's return was, directly, the government of Neville Chamberlain in Great Britain during the Second World War, when he tried to minimize the menace of the Nazi regime for political convenience."
(Admittedly J.K. Rowling is often somewhat less than believable as a source about her own stories, but this seems very plausible, as there are much-discussed parallels between Voldemort and Hitler, and both Fudge and Chamberlain were leaders who refused to acknowledge the imminent threat of war before being replaced by warlike ministers.)
So what happened to Chamberlain after war broke out? Well, he died naturally in 1940, a few months after leaving office, and there's no indication that this happened to Fudge; indeed, a year after his resignation he's certainly still alive for Dumbledore's funeral. However, the parallels between Hitler and Voldemort in terms of British governmental reaction rather break down in book 7, as Hitler never took over the British government, so we may be barking up the wrong tree here.
Answer partly inspired by alexwlchan's answer to a similar question on another SE site, but with other information as well.