It's a common theme in certain genres of poetry - I'm thinking of elegy, bucolic, Idyllic. Here's an example from Virgil:

Parthenius. Even now, methinks, I range O'er rocks, through echoing groves, and joy to launch Cydonian arrows from a Parthian bow.- As if my madness could find healing thus, Or that god soften at a mortal's grief! Now neither Hamadryads, no, nor songs Delight me more: ye woods, away with you! No pangs of ours can change him; not though we In the mid-frost should drink of Hebrus' stream, And in wet winters face Sithonian snows, Or, when the bark of the tall elm-tree bole Of drought is dying, should, under Cancer's Sign, In Aethiopian deserts drive our flocks. Love conquers all things; yield we too to love!"

If you have trouble following, the speaker is pining for an unrequited love; this is a list of things that he might do to catch the other boy's attention, and in the end resolves nothing works. That last line is where we get "love conquers all," but it means exactly the opposite thing people usually mean - it's not a happy ending. Love conquers all in a nasty, Roman way - Parthenius is basically saying "fine Love, you win. I quit."

Here's a bit from Ovid:

My mind’s gone, I’m dying, when you confess your crimes, and the blood runs cold in my whole body. Then I love, and hate, in vain, what I have to love: then I wish, with you, that I was dead!

Iphis in Ovid's Met committed suicide because of unrequited love.

On the Greek side Theocritus' first Idyll is about the shepherd Daphnis committing suicide over unrequited love.

Come master, and take this pipe... for Love calls me now to Hades... saying this, he made an end.

The singer in the third Idyll also contemplates suicide

I'll strip off my coat and leap into the waves from the place where Olpis the fisherman watches for tunny. If I kill myself you, at least, will be happy.... I'll lie here where I've fallen and let the wolves eat me.

Then there are the suicides of tragedy, most famously Antigone and Haemon. It's not clear why Antigone killed herself but Haemon seems to have done it for lost love.

In Euripides' Phoenician Woman, Menoeceus kills himself for the good of the state.

You can see unrequited love is a big driver for suicide, but these are all literary tropes. I don't know how representative they are for actual practice, but writers were clearly interested in the idea of teenage suicide.