What do you do if one of the users of your website announces that he will commit suicide?
You should handle it just like any other threat to someone's life. A person threatening suicide, or to kill someone else, is still threatening to kill someone. The only sensible thing to do is try and report it.
If you have the visitor's information, you may be able to get in touch with a local police department. This might be difficult if the visitor is from another country. If not, you have the FBI, etc, even if they tell you that there's nothing they or you can do.
If the person followed through, at least you tried to report it, furthermore, nobody can say you failed to report it, though I hate to bring up public perception in such a topic.
The biggest flashing red light with screaming alarms is when someone starts talking about how they are going to do it. If you ever hear or read about anyone talking about their own suicide in terms of how they would/are planning on executing it then you owe it to them and your conscience to try and get them help.
How to do that is answered elsewhere in the answers to these questions, but if you ever see someone talking about how they're going to do it, you have a moral obligation as a decent human not to ignore it.
There's probably not a lot you can do and I don't think that there is any right answer.
I was going to say that I would either take the post down or not let it through (depending on your moderation scheme), but that could be interpreted the wrong way by the poster. However, you don't really want such posts on your site - other users might not be sympathetic or tactful.
The best you can do is not be judgemental and reply with a link or phone number to the Samaritans (or equivalent) in their country (assuming you know the country).
Reddit has a "suicide watch" forum at http://www.reddit.com/r/SuicideWatch/ where users can go for help. You could point the user there, in addition to calling the police in his area.
Here is a list of international suicide hotline numbers:
I guess it really depends on the kind of website you have, the audience it caters to, and just how much personal information you have on users.
Suicide is a Big Thing, but 98% of people who threaten suicide just don't friggin do it, and are only uttering said threats out of a need for attention. Don't provide the attention, they'll go elsewhere. Give them chocolate candied attention, they'll pester you again, and again, and again.
PS: No, really, if you've got a suicide helpline of a website, or one targeting troubled persons (former cancer patients or something), yeah, call the cops if you have enough info. That might just be a real Desperate Cry for Help®.