Presumably, in Faulkner's "A Rose for Emily," Tobe stays with Emily until her death because he needs a job, or out of a mistaken sense of loyalty. But, really, that is just speculation. We can only guess.
Before Emily's death, the speaker really couldn't elaborate on why Tobe stays without ruining the surprise ending. And there's no reason to explain his staying until the body upstairs is revealed. There's nothing unusual about his staying until we find out about Homer.
He leaves in a hurry, of course, so he doesn't have to answer questions about the body or take responsibility for what's been going on in the house.
Once Emily dies, the only thing revealed is that Tobe skips out the back door. This, of course, heightens the suspense as the reader wonders why he so urgently leaves.
As far as is revealed, no, there is no love, at least not romantic love, between Emily and her servant.