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).
It has too low a temperature. To not emit much in the visible part of the spectrum, a star has to be very cold indeed, at most a few 100K, when radiation is mostly in the infrared (which is still detectable). By definition, a star is an object that undergoes (or has undergone) hydrogen burning (H$\to$He fusion). This sets an lower mass limit of 0.08M$_\odot$. All objects of that mass (or more) ever formed in the universe are still much hotter than 100K. However, in the very distant future, some 'dead' stars (without remaining fusion energy source) will cool down to such temperatures, this includes white dwarves.
It has too high a gravitational redshift for any light to appear in the visible. In fact, the so-called stellar-mass black holes (remnants of supernova explosions of massive stars) may in fact be such objects: strange stars denser than a nucleus (consisting of a quark-gluon plasma as dense as a neutron) but with such a small size and high mass that any radiation emitted from their surface is redshifted by a factor 1000 or larger.
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.