First, I think it should be established that while someone who is suicidal is more than likely depressed, self-harm is not necessarily bred of depression. Some people are just so terrible at coping with strong, negative emotions that they self-harm simply because they are upset in some way. It very often is linked with depression, but is not exclusive to it.
Back to the topic at hand.
Why can't you be in a romantic relationship with someone who is depressed or self-harming? The answer is pretty simple: they're not stable enough for it. Relationships are hard enough when both people are 100% mentally and emotionally healthy. If one of them isn't, things only degenerate faster.
Let's create a fictional couple, Victor and Millicent. Victor is the "normal" one - he's happy with his life and his choices, when he's upset he recovers quickly, etc. Millicent, however, suffers from depression, and not only has suicidal thoughts but also self-harms. While in many cases you may not know the other is suffering, in this one, let's say that Victor knows about this before he gets into a relationship with her.
When they get into the relationship, Victor is head-over-heels for Millicent. He knows that she's hurting, and he wants to be there for her in every way possible. He thinks being in a relationship could help with her depression; she knows his feelings are genuine, and he'll be able to help her cope with everything she's going through.
Millicent thinks the same thing. How could she possibly stay depressed when she has Vincent? She wants to be with him, he wants to be with her, and now that she has this love in her life, she'll be happy again.
Neither of them realizes that depression is so much BIGGER than they are. It isn't something you can rationalize away. It isn't something that you can convince yourself out of. Clinical depression requires medication. Plain old depression requires therapy.
The first time Millicent regresses, let's say it isn't anything to do with Vincent. They didn't get into a fight, they're doing just dandy; but still she falls back into depression and self-harming and wishing she were dead. Because she doesn't want to worry Vincent, she puts up a façade of happiness, and maybe she's good enough at it that he doesn't notice.
Then he walks in on her self-harming. He's upset, because he feels like he should have realized something was wrong. She's upset, because she couldn't be "good enough" for him. She couldn't be "happy enough". Now she's hurt him because of it, and that only intensifies her feelings of worthlessness and self-loathing. Vincent thinks he's failed her, and he redoubles his efforts to make her happy again.
Things improve a little. Millicent levels out again, although she occasionally returns to that dark place, though never for long. But nothing lasts forever, and they have their first real fight. Maybe it's over something stupid. Maybe it's not. It doesn't matter.
Now when she returns to depression and hurting herself, Vincent thinks it's HIS fault. He shouldn't have gotten upset with her. He should have let her have her way. If he had just been better, if he had just gotten over it, she would be fine.
Millicent feels doubly upset - not only is she hurting because of what happened with Vincent, but now she feels like maybe Vincent CAN'T help her. And part of her hates him for that, and really hates herself for not being able to love him enough to stop.
What neither of them realize is that it has nothing to do with Vincent. It has nothing to do with their relationship. Millicent has to stop because SHE wants to. She has to do it for her. Not for him. And as long as they're together while she's depressed, it's going to be a vicious cycle of anger and disappointment and hurt. It won't be fair to either of them.
This may seem... over-the-top to some of you. Like I don't understand you or your relationship with that person and hey married people get through this shit ALL THE TIME.
Here's the thing though: they don't. And even if they do, they more than likely were married before the onset of depression, so they have some semblance of normalcy to hold to in their relationship. When you enter a relationship with someone who's depressed or self-harming, there IS no normal beyond self-loathing, crushing apathy, and near-constant physical and/or mental pain.
A person with this sort of suffering simply cannot handle a relationship. Once they're better, and I mean gone to therapy, gotten medication, whatever it is that they need to come back to a plane of existence where everything isn't worldsuck, have the relationship of your lives. Until then, though, you have to back off. You can be there for them as a friend, but nothing more.
It sucks, I know. But it is so much better to wait until they're emotionally stable enough for a relationship, so that if God forbid something happens, you don't spend your life racked with guilt. Furthermore, if you're in love with this person, you shouldn't live with them or do anything else that makes it so that you're on a more intimate level. The same dangers apply.
All in all, if you get nothing else out of this post, just remember:
Even if you don't fully believe me, or think that your relationship will be different...
If you love that other person, don't take that chance.
Wait until he/she is better. It will give you a better chance at a lasting relationship.