In The Naked Now, Tasha and Data have a 'moment' together whilst under the effects of the polywater virus. Later on we see he still has a holo-statue of her. Clearly he has 'feelings' for her (of some kind), even after she had
Passed away.
Did their relationship ever continue, even if as just close friends?
It didn't continue romantically: at the end of The Naked Now, Tasha tells him, "I'm only going to say this once: It never happened", and as a result, when he is asked to explain why he has the holo-statue of her (in The Measure of a Man), he initially demurs.
As NominSim comments above, they were good friends at the time of Tasha's death, but it's never indicated that this is a result of, or continuation of, the events of The Naked Now. Most of the series' protagonists seem to become good friends early on.
It definitely continued as good friends, in her memorial speech Tasha calls Data "my friend". As far as canonical evidence towards their physical relationship continuing, I do not recall any.
As per this answer Data's last known intimacy that he talks to the Borg queen about, is likely Tasha, however the star dates given by the writers were likely mixed up (as it is a date after she died).
There is most likely, though I refuse to research it probably plenty of "fanfic" dedicated to this subject, however as far as straight from the canonical sources, I do not believe anything is said about their physical relationship continuing, in fact Tasha I believe remarks to Data that "it never happened" at the end of the episode in question. That coupled with her speaking of him as a "friend" in her memorial speech, puts him squarely in the "friend-zone" (obligatory TV-Tropes warning) I am afraid.
