HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

In the regular version, the elixir doesn't heal anything. After each round, it should fully restore HP, but it doesn't. Is this a bug in the game or just in my specific copy? Has it been fixed in the HD mod?

I would assume this is a bug caused by user error. For example, the elixir might be equipped in a disassembled state. Or it might not be an elixir at all, but a wizard's well. Or something else.

In any case, bug reports should be accompanied by a save file demonstrating the issue.