I know that a teacher should report to someone if they think that a student is having suicidal thoughts. But is there a law in New York State that requires it?
This is a difficult question that does not have a clear answer.
There is a duty to report suspected child abuse or neglect, but suicidal tendencies don't necessarily flow from either of these causes.
There are also duties of a teacher in loco parentis as the person temporarily having custody of a child from which one could argue that this duty arises, as well as a duty not to abuse or neglect a child that is in your custody.
But, what constitutes acts sufficient to trigger this obligation if it is not imminent (e.g. the child trying to jump out of a fifth story window, or cut an artery while class is in session), is very muddy.
Also, in a public school, there would also be an additional layer of governmental immunity of a teacher from liability unless it is expressly waived by statute.