It happens in both Survival and Creative.

Touchscreen is turned off. I have Sticky keys disabled, if I mess with that, it makes the inventory not able to touch anything at all but if it's off, then I can pick it up for a micro-second, and it'll be placed back into whatever spot it was in. It won't let me place anything in chests, it'll disappear for a second and then re-appear to how it was before.

Read MC-5130 and the others but it doesn't not apply since TouchScreen is already turned off. I went onto two other servers and I do not have a problem there. I do not know how to re-create the problem there as I was making some fireworks just fine (in creative) and then I started to started to make some more and suddenly, can't do anything. A player mentioned sticky keys but I disabled them ages ago.

I forced a crash and am including the crash report just in case you see anything that might help.

Edit: I also tried re-installing Minecraft and still just doesn't let me move inventory stuff in the one server, the other servers are unaffected.