Can the Sharingan copy Genjutsu in Naruto?

 Yes.

Functionally, copying genjutsu is no different than copying ninjutsu or taijutsu.

Taking advantage of the Sharingan’s Eye of Insight and Eye of Hypnosis, the copyist can effectively learn any technique by memorizing another’s hand-seals while simultaneously watching the flow of chakra throughout their body.

Demonic Illusion: Mirror Heaven and Earth Change is based of this very concept:

Kurenai wove her hand-seals in plain sight of Itachi’s Sharingan. Consequently, Itachi was able to copy and cast the same genjutsu on Kurenai — in turn reversing her own:

Nevertheless, while most genjutsu can be easily copied by the Sharingan, there may be a few requisites required to use them. Case in point, Shee’s Lightning Release: Lightning Illusion Flash of Lightning Pillar:

If the copyist is unable to use Lightning Release, he’ll be able to dispel the genjutsu, but he won’t be able reverse it back onto the caster.