So there are a couple different categories of abilities that I'm wondering about. They may all act the same, but I think I've noticed them behave differently in game.
DoT effects? I assume Teemo's poison will continue to tick even if Teemo dies.
Anivia's Wall? Does it last the entire duration regardless of Anivia being alive or not? Or does it disappear on death?
What about Demacian Standard cast right before death by J4? Does that continue to apply the team buff for the duration or is it gone when he dies?
Gragas' Barrels and Lux's Lucent Singularity? I've seen these disappear at death and hang around after the caster's death. My theory is that they will stick around if they have stopped moving by the time the caster has died.