The relationship between Gaz and GIR is somewhat similar to Gaz's Relationship with Zim in that she hardly has any interaction with the insane S.I.R Unit out of hate with any physical contact with just about anyone.
Their limited interaction however is quite predictable (and somewhat different) in that GIR seems to be infatuated with Gaz but she has little tolerance for him, which surprisingly GIR seems to understand as hinted in "Bloaty's Pizza Hog".
But unlike anyone else, she actually gives a warning rather than any type of physical or verbal abuse, probably because she knows GIR is too stupid to understand the full extent of the harm she could give his way.
During "Tak: The Hideous New Girl", they interacted yet again but only because there were no video games in Zim's Base and was forced to team up with the malfunctioning SIR Unit to help save the Earth, amazingly GIR was able to make Gaz dance and even went as far as to give her a kiss much to her disgust.
In their final interaction in "The Voting of the Doomed", she ate the salted nuts straight from his head making it seem like she was eating his brain which GIR didn't seem to notice or mind.
It is unknown how they would interact had the series finale movie "Invader Dib" had been made since one of the plots presumably involved Gaz teaming up with Zim and becoming Irk's co-ruler, presumably becoming GIR's "mistress" as a consequence.
Invader Zim (comic series)
In Issue 21, after the four-way body swap among Gaz, GIR, Zim, and Dib, Gaz apparently employs GIR. He's the only one she allows into her dressing room, and after she takes over the world and exiles Zim and Dib to the moon, she keeps GIR around on Earth. It's worth noticing that Gaz treated GIR better than in the series, although this might be in part due to the swap.
In Issue 38, after GIR starts impersonating Dib, Gaz immediately sees through his poor disguise. She then convinces him that to properly carry off the impersonation, he has to try and prove that Zim is an alien. This causes GIR to really get into his role, to the point that it ends up ruining Zim's latest plan.