Hey LadyKay,
That kind of of emotional blackmail is rough to deal with, and is a sign of a very toxic relationship. You really need to set personal limits and avoid being put into this situation. You can do this by expressing support or concern, but you need to put the choice for life or death right back where it belongs, with the person making the manipulative threat.
Do not fight with this person, don't give in to their threats or it will just create a pattern of manipulation, and don't accuse them of manipulating you. Giving into threats like this won't solve anything. You will still feel manipulated and angry and they are still at risk of harming themselves.
I don't know if this is a relationship with a family member, a friend, or a partner, but regardless, this person isn't not currently well enough to be in any kind of relationship and needs to get better. If they are serious in their threats, they need help beyond what anyone on a bulletin board like this can give for advice. Maybe try calling a crisis line and ask for guidance. Or if you work for a place that offers a few sessions of limited Councling for yourself, like an employee assistant program(and quite a few company's have this) that may help both you and the person your dealing with. I know I'm certainly not a professional and wouldn't want the responsibility of giving poor advice for your circumstance.
Its been many many many years since I've been serious on this board. Even back when I posted regularly. I've been in your situation and I also know people who have, so I can completely empathize with you, and felt I should respond to your post. I know your situation sucks. Get some real help. Even if it's just a suicide prevention hotline.
And now I think I would like to now go back to lurking and only playing the fluffy games in the introduce yourself section and periodically checking the nightlife for what's happening please. (But seriously, shoot me a message if I can help or you need someone to talk to.)