When the PS Store closes on PS4, what will happen to the library?

That will be years before it happens.

Like the other guy had stated, it won’t be until at least the end of this decade before that will likely happen. By then we should be at PS6.

But with that said, I think chances are, it’ll likely be the same idea where you can still download your purchase but just can’t buy things anymore.

It's going to be years before the ps store closes on ps4. If you look at the data, the ps3 store is closing approximately 15 years later, and for now purchased games can be re-downloaded. So on ps4 that would estimate around 2028. But the circumstances are different then before, PS4 is significantly more popular then PS3 was, espically the digital sales aspect. Also assuming Sony sticks with x86-64 for future generation playstation, its likely that ps4 compatibility will remain, which gives an incentive for Sony to preserve the PS4 games on the store. Eventually by the time the servers shut down for both the store and re-downloads, ps4 will be irrelevant and you will be able to resort to the internet archives of games to play on hacked consoles or emulators. Since that community always will preserve the games.. 

Yes, your purchased items for PS3 will still be available to download or re-download. You won't be able to buy new PS3 content, and your wallet will only be good for PS4 and PS5 content. (You'll be able to request a refund if you only have a PS3.)