What happens if the title owner rejects to fight anymore or announces that he/she is retired? Is a new belt made? Should he/she return it?
In boxing, the fighter who retires will keep the belt and the committee will generally schedule a fight between the next two contenders to recognize the new champion. A new belt will be made for the new champion.
Lennox Lewis was a recent retired champ.
In terms of belts a new belt is made every time a new champion is crowned. The ex-champion gets to keep his belt. (source: Wikipedia)
Champions maintain permanent possession of these belts even upon losing their title, with a new belt made when a new champion is crowned.
I boxed for a couple of years in the 90s and spent some time training in Colorado Springs and San Diego - the hubs of USA amateur boxing at the time (alas the furthest I made it was GG Finals - lost to a damn lefty). I knew some "Champions" and there were tons of federations at the time. The question about the belt is easy. The belt is yours. I mean there might be some really small federations that make you take the belt to fight and hand it over but none that I knew of. Most of the belts that were shown off to me were not "active". And I can swear I saw some show similar to "How it's Made" that went to a factory that made these belts.
Really there isn't a retirement process in boxing. You can say you are retiring but in essence you are vacating the belt. Basically you do not meet the requirements to defend the title for that federation. All federations have a minimum on the amount of time between fights. Most even make you box someone within a certain ranking range.
So many constitute that a title must be defended every 4-8 months (differs widely) and that you must fight someone within top 20-50th ranked. Most fighters don't really "retire" because they might as well wait until the last minute in case they want to fight or need the money. How the federations handle this is what you would expect - well it is drastically better than it was. They can get a scrub properly ranked to meet needs or give someone medical time to prolong there retirement or whatever. They make the rules. They only care when it comes to money.
Once someone has fully vacated the belt how the federation handles setting up a new champion has to do with their bylaws, money, and randomness. Often promoters vie to get their people the title shot. Federations are often pulled in 10 different directions.