I had read that it was possible, but after many, many years I had never experienced it myself. Now I have just managed to die in Monkey Island 1 (and I'm not talking about the rubber tree joke). Since it implies either not knowing what to do for a long time or actively pursuing it, I had never seen this "Easter egg".

Mild spoilers

The way to die is by drowning. Earlier in the game Guybrush brags about being able to hold his breath for 10 minutes. Once the sheriff throws you underwater after stealing the idol, if you don't escape after 10 minutes, Guybrush will actually die!

https://www.gog.com/upload/forum/2020/04/c051914caedd003661aca5c7b4b68ce32979a9c4.png

This whole scene is hilarous. I love as well the brief dialogue that takes place around 3 minutes in between the criminal and his friend where it seems that you are going to be magically given a knife to cut the rope. :D