Common perceptions of mental illness and relationships suggest that mentally ill people do not belong in relationships, do not deserve love and affection, and are even dangerous to marry or get involved with. Not for nothing are undesirable prospective partners 'crazy bitches,' are former partners whom we're supposed to hate 'crazy exes.' It is highly unusual to see a depiction of a functional relationship with a mentally ill partner or partners; such a thing is alien to the arbiters of relationships in pop culture.
Some depictions of mental illness in pop culture suggest that we are all overflowing with libido, unable to exercise any degree of control or restraint when it comes to sexuality. The oversexed female character in particular is a very familiar stereotype; look at Brenda on Six Feet Under, who is introduced to us having sex in an airport closet. On the road with Nate to give him a ride home, she says they shouldn't pretend this is the first time either one of them has had sex with a stranger. Implication: Stranger sex is just a thing that she does, and over the course of the series, it becomes apparent that this is because of her mental illness.
Mentally ill people are also boyfriend stealers, particularly on teen dramas. Unable to have relationships of their own, they horn in (so to speak) on existing ones, without a care to the impact that it might have on their friends. Again, these characters are often depicted as out of control and scary; it is not uncommon to have a boyfriend stealer turn into a stalker, for instance, relentlessly refusing to give up after a one night stand.
Or mentally ill characters are abusive, and it is very much attributed to their illness within the context of the story. Commonly the mentally ill partner in this context is male, and uses both physical and emotional violence against his partner. Mentally ill men, pop culture tells us, are either abusive and terrifying or oddly brilliant and inspired by their illness. (See Billy, again on Six Feet Under, who is miraculously creative as a result of his bipolar disorder, except when he's in the height of a manic phase and he's running around doing things like trying to knife people and trashing offices.)
They can also be controlling and manipulative. Terri on Glee is one example—she doesn't let Will have a life and tries to exert as much control as possible over everything, right down to forcing Will to do things he doesn't want to do. Such partners become impossible to escape because they are so obsessive, and they drag their other halves down. Create a burden.
In all cases, mental illness becomes an object of resentment and hatred, both within the text and for viewers. One of the fastest ways to code a romantic partner or love interest as inappropriate or undesirable is to make that character mentally ill, or to give that character vague symptoms of unspecified mental illness. Viewers are encouraged to hate such characters.
They make ideal foils for emotions viewers may not feel comfortable expressing as well; they don't hate her because she engages in stereotypically feminine behavior, but because she's mentally ill, say. It's notable that many of the traits associated with mental illness in pop culture, like being self-centered and controlling, are also associated with women in misogynistic narratives.
Free to hate these characters, consumers of pop culture can hope that the hero eventually escapes and manages to date the real love interest, the nice, normal, fun character who is enjoyable to be around. This plays out in real life as well, of course; how many people are told to leave partners who behave erratically or are open about living with mental illness? How many "friends" secretly hope that so and so leaves her "annoying" partner because she's just too unstable to invite to dinner parties?
Issues like controlling behavior, selfishness, abuse, and incompatibility can come up in all relationships, not just those involving mentally ill people. And they are all good reasons to leave relationships; no one should be forced to stay in a dangerous or unhappy relationship. But, at the same time, when these things are always attributed to mental illness, it suggests that relationships with mentally ill people will be inherently dangerous or unpleasant, that mentally ill people have nothing to bring to relationships, even that mentally ill people should not be allowed to have relationships.
The stereotypes about mental illness and relationships that appear in pop culture have very real implications for actual people with mental illness. Many of us navigate relationships just fine every day. Many more of us are actually at increased risk for abuse; emotional, physical, financial, because mental illness can make us vulnerable. And yet, we don't see depictions of rape of mentally ill women, for example, or financial abuse of mentally ill men by "caregivers." These very real issues are not often depicted in pop culture.