Is there any reference to say where he went or what he did after leaving his post? What actually happened during his life from that point?
In Half-Blood Prince, when Fudge and Scrimgeour visit the Muggle Prime Minister, we learn that he’s still working as an envoy to the Muggles.
I am not aware of any canon sources that describe what happened to him after this.
Specifically, I do not know of anything in the books, films, a J.K. Rowling interview or Pottermore update that explains what happened to Fudge. We see him briefly at Dumbledore’s funeral, but nothing is said of what he’s doing.
That makes for a rather boring answer, so a little speculation.
I don’t think he would have stayed on as envoy to the Muggles. Voldemort has no need for such a person in a post-Scrimgeour Ministry. I also don’t think the Order would have trusted him, so he’d probably be on his own.
His final year at the Ministry – denying Voldemort’s return – was almost certainly a big help to Voldemort, so I don’t think he’d get murdered. He also wasn’t a particularly important target, and wouldn’t merit Death Eater attention. I think he retired and fled the country, and probably kept his head low.
What about after the War? Dumbledore hints at one possibility:
“I tell you now — take the steps I have suggested, and you will be remembered, in office or out, as one of the bravest and greatest Ministers of Magic we have ever known. Fail to act — and history will remember you as the man who stepped aside and allowed Voldemort a second chance to destroy the world we have tried to rebuild!”
— Goblet of Fire, chapter 36 (The Parting of the Ways)
We know that Fudge, and more generally the Second War, was based upon Chamberlain’s Government:
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.”
I think what happened to Fudge is the same as what happened to Chamberlain. In World War Two, Chamberlain tried to appease Hitler, and resigned when Hitler made his true intentions clear. Chamberlain also stayed on as a member of Churchill’s war cabinet, not unlike Fudge’s role in Scrimgeour’s Ministry.
Chamberlain died of cancer shortly after he resigned, but got a positive eulogy from Churchill:
Whatever else history may or may not say about these terrible, tremendous years, we can be sure that Neville Chamberlain acted with perfect sincerity according to his lights and strove to the utmost of his capacity and authority, which were powerful, to save the world from the awful, devastating struggle in which we are now engaged. This alone will stand him in good stead as far as what is called the verdict of history is concerned.
(In this analogy, Dumbledore corresponds to Churchill, so falls apart here.)
Fudge would never be remembered as well as Chamberlain. Both of them tried to deny the possibility of war, but Fudge tried to discredit both Harry and Dumbledore while doing so. That’s more than just burying your head in the sand.
I think he would have retired in relative obscurity, and just as Chamberlain is synonymous with appeasement, so Fudge would become synonymous with criminal blindness, and actively fighting those who were trying to tell him the truth. I’m sure he’d be disappointed, but it’s what he deserves.
My alternative theory is that he went to start a chocolate factory.
We know that he stayed on and helped the next Minister for a while at least, since he introduced him to the muggle prime minister. Otherwise I don't believe they bring him up after that scene. He wasn't a notable at the last fight in book 7 so I think we can just assume that like our typical presidents/prime minister he retired/gets a pension, OR he was a casualty of the war as a high profile figure.
His Wikia page states:
He stayed on as an adviser and messenger to the Muggle Prime Minister for Scrimgeour, as Scrimgeour was too busy to contact the Muggle Minister himself.
It is unknown what happened to Fudge after Lord Voldemort took over the Ministry of Magic, but he lost his position as messenger between the Minister and the Muggle Prime Minister, as the Death Eaters had no use for Muggle alliances.