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:
We know that Fudge, and more generally the Second War, was based upon Chamberlain’s Government:
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:
(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: