I'm of the opinion that the threads should be locked and members who make them should be reached out to privately with resources and support from the mod staff and/or other trained volunteer users. A lot of the reasons match things that
TheLostBigBoss and
Ultima_5 have been saying, but there's another perspective worth considering:
When a suicide thread stays open, you're making unwitting helpline volunteers out of the entire forum. Every time I see one of them I feel a sense of responsibility to check it, see if there are any updates from the OP, and try to post something encouraging if necessary out of a desire not to have it on my conscience that I could have done something. What happens to the users like me when we do our best to reach out and help someone obviously desperate, and then we never see another post from them? What if that happens repeatedly? There's a real risk of triggering very harmful reactions. This stuff isn't something I would have signed on for voluntarily, but now I'm unable to extricate myself from it due to my sense of obligation to the community.