Question

if someone moves me while I'm using the Exaltation Forge do I lose items? 

I have a friend who me that he lost the items that way

Answer 1

No, you wouldn't lose any items.

Just like using a Reward Shrine, you can open the menu for it and another player move you around by pushing you, however you can still click buttons in the menu and they will work as they should.

Personally I've got some slivers from the forge while being pushed and it worked normally however if that's not the case your friend should open a support ticket reporting the situation as that could've been a bug.