Once an axe has been used and durability is decreased it is unable to be removed from Hotbar..
This happens with both diamond and iron. I have not tried other axes..
Logging out and back in fixes it.
It's just like the issue when boots with soul speed cannot be removed from player once durability goes down.

How to recreate:
Get an axe in your hotbar
Chop down a tree
Axe is now stuck in your hotbar