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.
This ticket is invalid as it relates to a modded or 3rd party client/server.
Any non-standard client/server build needs to be taken up with the appropriate team, not Mojang.
Any plugin issues need to be addressed to the plugin creator.
This site is for addressing issues related to the base unmodded Minecraft.