In Faulkner's "A Rose for Emily," Tobe was "an old man-servant – a combined gardener and cook." More, he was the first one to know about her and her father's death, and the only one to know about Homer's death. He may not have had access to Emily's bedroom, her inner sanctum, but he definitely could deduce the source and circumstances surrounding the smell from it.
Tobe and Emily trusted each other, and he may have been the only confidant she had. Certainly, he's the closest thing to a husband that she ever had, even though they were never romantically involved. As such, Tobe leaves her premises once the white townspeople arrive on the crime scene. He has protected her privacy for years, and now he does not want to be a witness who divulges gossip. In this way, he is a foil for the narrators. Whereas they divulge all they know, even misinformation and hearsay, Tobe divulges nothing: he is as silent as a corpse.
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.
I think that Tobe stays with Miss Emily because it is the only life he has known. He has surely been with her family for a long time. I think that he is a symbol of the old days in Southern society where a black servant or slave belonged with a given family and came to be identified with them.
I think he leaves for two reasons. First, he was Miss Emily's servant and that was all he seemed to care about (as far as white society went). Once she was dead, he had no reason to stick around the white people.
Second, I think that Faulkner has him leave to emphasize just how cut off from society Emily was. She was so alone that even her servant had no connection to the rest of society (or at least white society).