Nathaniel Green has written an excellent article, World Champions Who Gave Up The Crown, for Chess.com which comprehensively answers the question. I'll try and summarise.
Garry Kasparov (1993)
Kasparov dropped out of FIDE's championship and started his own via the PCA.
Anatoly Karpov (1999)
This was completely new to me. I hadn't realised that FIDE had made such a dog's breakfast of the world championship that Karpov at a certain stage just withdrew. Here's the explanation from the article:
FIDE almost immediately started messing with the world championship format. In 1996, Karpov was seeded into the semifinal of a 13-player match tournament and defeated GMs Boris Gelfand and Gata Kamsky to keep his title. Karpov's 1998 defense was even less traditional. There was a year less to wait and the Candidates process involved a 97-player super knockout tournament, although Karpov was in the final this time.
GM Viswanathan Anand won the knockout and then had to start playing Karpov less than a week later in a six-game match. Karpov won in rapid tiebreaks. One of his easier championship paths, however, was going to have to be succeeded by his most difficult yet.
The very next year, that super knockout tournament was to be the world championship itself, and Karpov was only to get a first-round bye like several other top players of the time. Karpov didn't agree to being the first champion to have to defend his title this way, so he simply didn't play.
Bobby Fischer (1975)
Of course, we all know Fischer declined to play Karpov in 1975.
Mikhail Botvinnik (1966)
Not quite the same thing. Botvinnik lost the title (for the 3rd and final time) and then declined to take his allotted place in the candidates. FIDE changed the rules, depriving the beaten champion of the right to an immediate replay. As Green puts it:
[Botvinnik], like Carlsen, was unhappy with the world championship process and thus simply stopped participating.
Alexander Alekhine (1946)
War and then death resulted in Alekhine losing the title. He was due to play Botvinnik but died before he got the chance. As the article notes:
Alexander Alekhine didn't exactly give up the title; he died while holding it. That makes it pretty hard to compare his situation to anyone else's, let alone Carlsen's.
...
Just as World War I wiped out a match between Lasker and Akiba Rubinstein, World War II prevented Alekhine and Paul Keres from playing for the title. This time Capablanca's role as the postwar challenger was filled by Mikhail Botvinnik. But, while trying to arrange a match with him, Alekhine was gone, less than a year after the war ended.
Emanuel Lasker (1921)
According to the article:
Emanuel Lasker was the first world champion to express disdain for the title and show, in a public way, that not everyone enjoys occupying the position.
...
Negotiations for the first world championship after the war, between Lasker and Jose Capablanca, grew as strained and bitter as store-brand plain Greek yogurt.
It reached the point where Lasker announced he was resigning his title and giving it to Capablanca. He would still play the match, just as the challenger. Nevertheless, he did, like Carlsen, want to give up the title.
