When a creature is invisible but not hidden, one could argue that the players still know it's general location.
If a creature becomes invisible and not hidden does it remain on the battle-mat?
An invisible creature is not hidden. All creatures are fully aware of which square/location it is in, therefore it remains on the battle mat.
This is one of the core Rules of Hidden Club.
The relevant rules are contained in the "Invisible creatures and stealth" definition.
Yep, remains on the map. Here are the 4e Official rules on Invisible.
Invisible Creatures and Stealth: If an invisible creature is HIDDEN from you, you can neither hear nor see it, and you have to guess what space it occupies. If an invisible creature is not hidden from you, you can hear it or sense some other sign of its presence and therefore know what space it occupies, although you still can’t see it.
Make a Perception Check: On your turn, you can make a Perception check as a minor action to try to determine the location of an invisible creature that is HIDDEN from you.