She’ll just keep on talking about random things until either she runs out of things to say or you get mad/bored/annoyed and decide to delete her. Act 3 is pretty much just a Boss battle against Monika. So basically, everyone dies and the protagonist is left to deal with the fact that basically everyone he knows is dead. Everything goes back to normal except there isn’t a Monika and Sayori is the new president of the club, only for her to become obsessed with you. What happens if you delete Monika after sayori dies?
It reaches a point where, after she confesses her love to the protagonist, she stabs herself to death out of over-excitement or rejection (depending on the protagonist’s response to her confession, both lead to the same outcome, courtesy of Monika). She is ultimately unable to control her obsessive outbursts. Even in the game, Sayori says she’s been dealing with this her whole life.) Why did Yuri kill herself? In her case it probably just developed over time, since worrying about college isn’t really enough to just ‘give’ you depression.
Sayori mentioned that she had suffered from depression for her whole life. You might be interested: How To Understand Literature? Why is sayori depressed?