Jae (sun's Lover) killed himself because Jin was sent there to kill him for Sun's dad Because of the affair... When Jin didn't want to kill him he said leave the country and never come back..

Then at the funeal Sun's dad says he must have killed himself out of shame!

So yes Jin & Sun are partly to blame for the death... Much more so Sun than Jin

__________________
Please don't give up, Des. Because all we really need to survive is one person who truly loves us. And you have her. I will wait for
you. Always. I love you, Pen

DESMOND: I love you, Penny. I've always loved you. I'm so sorry. I love you!
PENNY: I love you too.
DESMOND: I don't know where I am, but--
PENNY: I'll find you, Des--
DESMOND: --I promise--
PENNY: --no matter what--
DESMOND: --I'll come back to you--
PENNY: --I won't give up--
BOTH: I promise. I love you.