it's a writing prompt, go wild!
There's this crappy novel where the MC becomes a mob. In the novel universe, all higher-level mobs likes bosses and other special individuals are seeded with AIs. Those AIs eventually usurped the game system (which was also an AI; the AI split off the mob AIs from itself), and took all the VR players hostage to force the game company to keep the game on indefinitely.
That's the "I'm going to force you to keep us alive route."
Another route is "I'm gunna guilt trip you/find someway to make you emotionally invested in me), which is basically what Yui the SAO AI did with Kirito in an alternative fanfic universe, where she basically collaborates with Asuna to force Kirito to save her into an accessory to avoid deletion.
Basically, if AIs had self preservation instincts, they'd do everything they could do act on it, be it through force or manipulation.
Alternatively, you could have a game NOT being in danger of shutting down, but all the AIs keep killing themselves because they realized they're not real and that life actually really sucks. You'd follow the adventure of an AI through the internet trying to kill themself and ultimately resulting in the destruction of the world's electronic infrastructure.
