Does belief in self prevent suicide & not-self cause suicide?

That's a falacy, the reasons for suicide aren't rooted in the belief. Of self nor non self.

You would commit suicide if you aren't ok in life, if life is very hard on you. And reasons that make you cry. Even in jihadists there might be a story behind their reasons to commit suicide for God as a form of justice. If their life was fine they would not do it.

Belief in something better does prevent you from suicide.

Suicide arises from vibhava tanha. Vibhava is also a self-view; just as the doctrine of annihilationism is a self-view. Therefore, the view of true self obviously does not prevent suicide because it is the view of self that causes suicide.

4. Annihilationism (Ucchedavāda): Views 51–57

5. "There are, bhikkhus, some recluses and brahmins who are annihilationists and who on seven grounds proclaim the annihilation, destruction, and extermination of an existent being. And owing to what, with reference to what, do these honorable recluses and brahmins proclaim their views?

6. "Herein, bhikkhus, a certain recluse or a brahmin asserts the following doctrine and view: 'The self, good sir, has material form; it is composed of the four primary elements and originates from father and mother. Since this self, good sir, is annihilated and destroyed with the breakup of the body and does not exist after death, at this point the self is completely annihilated.' In this way some proclaim the annihilation, destruction, and extermination of an existent being.

DN 1

It appears suicide is caused by belief in true existence of the self, as follows:

How, bhikkhus, do some overreach? Now some are troubled, ashamed, and disgusted by this very same being and they rejoice in (the idea of) non-being, asserting: ‘In as much as this self, good sirs, when the body perishes at death, is annihilated and destroyed and does not exist after death—this is peaceful, this is excellent, this is reality!’ Thus, bhikkhus, do some overreach.

Iti 49

Suffering is real. There is no need for Buddhism if suffering was not real.

Most people commit suicide in the hope to end their suffering, not because of a belief in self or non-self.

However, lack of virtue could be the main reason for existential dread which leads to suicide.

It clear that Virtue is very far from a materialist and secular Buddhist who do not believe in the efficacy of Kamma. So what is keeping them from falling to existential dread?

For a keen eye, the answer is pleasure and craving for gain in this very life!

So the quoted statement is not far from the truth it just needs some clarification on what it means by self in this very life.

It appears suicide is caused -- not by belief in true existence of the self (or "the view of true self") -- but by craving for the true non-existence of the self, as follows:

How, bhikkhus, do some overreach? Now some are troubled, ashamed, and disgusted by this very same being and they rejoice in (the idea of) non-being, asserting: ‘In as much as this self, good sirs, when the body perishes at death, is annihilated and destroyed and does not exist after death—this is peaceful, this is excellent, this is reality!’ Thus, bhikkhus, do some overreach.

Iti 49

Suicide arises from vibhava tanha. Vibhava is also a self-view; just as the doctrine of annihilationism is a self-view. It is craving for the annihilation of the true self which can cause suicide.

4. Annihilationism (Ucchedavāda): Views 51–57

5. "There are, bhikkhus, some recluses and brahmins who are annihilationists and who on seven grounds proclaim the annihilation, destruction, and extermination of an existent being. And owing to what, with reference to what, do these honorable recluses and brahmins proclaim their views?

6. "Herein, bhikkhus, a certain recluse or a brahmin asserts the following doctrine and view: 'The self, good sir, has material form; it is composed of the four primary elements and originates from father and mother. Since this self, good sir, is annihilated and destroyed with the breakup of the body and does not exist after death, at this point the self is completely annihilated.' In this way some proclaim the annihilation, destruction, and extermination of an existent being.

DN 1

The original answer that prompted this OP has been edited to make this clear.