The same thing that happens to any other five figure asset you own. It goes through whatever the probate process is wherever you live, which can vary a whole lot.
I plan to sell my contracts when my kid is in high school, but if I were planning for them to be inherited, I would talk this through with my estate attorney. There are a lot of ways to structure this.
It all depends how you have your deed recorded. If it’s just in your name it becomes part of your estate. I suppose you could include it in your will. If it is in multiple names it goes to the survivors.
If you have a deed that names both you and your spouse, or others, in a way that shows it is as joint owners with right of survivorship, then the death of any one owner automatically results in the remaining owners having complete ownership. The property will not be subject to probate as long as any one of the owners is still living, although an estate tax filing for the deceased may be required (having to make a filing does not mean there is a tax since, in most cases, only the very rich will see a tax).
With that type of joint ownership, the issue of probate does not arise until all owners have died. And the probate of the last owner will not just be one that is part of the probate procedure in the home state of the deceased but a separate one that has to be filed and completed in Florida (or other location of the DVC Resort), a procedure that can cost real money, including for the need to hire a local attorney.
If the last owner has a will covering what is to be done with the property, then the property will be transferred as set out in the will or possibly even sold if the named recipient disclaims the inheritance (a possibility if the named recipient does not want to take ownership for any reason, e.g., does not want the expense of paying dues). If there is no will and nothing else, like a trust, determining what happens upon death of the last owner, the property will be distributed in accordance with the applicable laws of the state in which the deceased was a resident.
Other than just a will, there are owners who set up trusts to cover the distribution of their property upon death. A common method is called a "living trust" under which the owner essentially retains all powers of ownership until death but the DVC property (and any other property put into the trust) is actually legally owned by the trust and a deed is filed naming the living trust as the owner. In those situations, the DVC member is deemed the "beneficial" owner and continues to be the member on the DVC account. A benefit of having such a living trust is that all probate involving the property is avoided upon death of the last owner because the legal owner of the property is the trust, which cannot die (although estate tax filings for the deceased can still be required).
Since we are talking about IF and not WHEN, the scary thought is what if I don’t die? Contract would expire and I’d balk at what ever ridiculous price the new ones are going for. It would be the timeshare version of “I remember when coffee at McDonald’s was 10c”!