Are there stars that do not emit any light in the visible part of the EM spectrum?
There are two possible reasons why a star would not be detectable in the visible part of the spectrum (even with the most powerful telescopes not yet invented), apart from the trivial (too far away, hidden behind screens of dust).
So if you consider those (still hypothetical) strange stars, then these are possible candidates.
Maybe a very old Neutron Star?
A black dwarf would not emit any visible light, but the universe is not old enough for that. Even the oldest and coolest white dwarfs still have a temperature between 2500-4000K (sorry for not remembering the reference for this).
Brown dwarfs (or planemos/sub-brown dwarfs) like WISE 0855–0714 could be as cool as ice. but they don't count as stars anyway.