What happens to components which stop being manufactured?

The designs are still the property of the original owners, and they don't generally "release" them. If there were licensed or unofficial 2nd sources it's more likely that was initiated when the product was more active.

I believe that sometimes a third party will buy up a quantity of dice and package them up as orders arrive. Running a few extra wafers at the twilight of the design life of a part may not be all that expensive. We buy a relatively expensive (some thousands of dollars) extremely specialized part where the original manufacturer has shut down their entire fab after making enough stock of bare dice for the foreseeable future.

Also, grey market dealers may hoard last-buy and surplus parts, as well as parts removed from scrapped PCBs. These have dubious provenance at best, and may be counterfeit or fraudulent parts at worst, but they may be a last resort. Some of the dealers are much more reputable than others.

Why would anyone want to buy old inefficient obsolete parts? Sometimes the cost of re-design or certification is so high that the end product will otherwise have to be abandoned. For example, when microcontrollers are obsoleted it can cause much gnashing of teeth, especially among smaller customers. This is particularly true when there is a technology and architecture shift such that there are no longer even somewhat compatible parts being manufactured.

Not usually, no.

Historically, many chip designs have been licensed to alternative manufacturers (a system called "second sourcing") because a lot of purchasers would refuse to buy a chip only available from one supplier. In that case, even after the original manufacturer has stopped a second source supplier may continue to produce the chip -- for example, this has happened to the Intel 8086 and 8088, which are still produced by the second source supplier Intersil even though Intel stopped producing them some time ago. But this isn't automatic, and depends on the second source supplier thinking there is still a market. It's also less likely to happen today because second sources are no longer considered as important, so many chips today only have a single supplier.

They are not "released" per se...

While maskwork and patent rights are of shorter duration than copyright protection, even the 10 year term of mask work rights is longer than the lifecycle of some ICs in this day and age. Patent protection is another issue -- the 20 year lifespan and the strong barrier to independent redevelopment that a patent provides means that risking a patent violation to redevelop an obsoleted function is simply not worth it in 99% of the cases.

However, there are options

In some cases, a part that has been obsoleted by its original manufacturer may have been second sourced, where a second or even third manufacturer has produced the part under a license of patent and maskwork (or just patent) rights from the first manufacturer. This was more common in the past for "commodity" analog, mixed-signal, and interfacing functions; while not used as much nowadays on new parts, some parts have become household names (think MAX232) partly due to widespread multiple-sourcing.

In fact, the original MAX232 design and mask are technically in the public domain now as all rights to them have expired. Maxim still makes the chip, though, in addition to several second-source suppliers, who would now be free to sublet production or revise the masks and circuit to their tastes within constraints of the license contract, or make new derivatives with different part numbers, even.

There are also parts that are multiple-sourced subject to industry standards, such as generic discrete semiconductors, common SSI/MSI logic functions, and memories small and large, which are generally covered by JEDEC standards such that any vendor's part may be used in any application that is designed to accept a part that meets specification.

In addition to second-sources or standardized sources outliving the original, there are specialists, most notably Rochester Electronics, that buy up packaged parts and/or wafers (during a "Last Time Buy" phase) and stockpile them for future distribution. In some select cases, obsolete-parts specialists such as Rochester may buy masks from an original or second-source manufacturer in addition to their Last Time Buy of produced parts, wafers, or dice -- this gives them the power to restart manufacture of the part at any time, subject to foundry constraints, as many obsolete parts rely on obsolete chipmaking processes that may no longer be in use.