In Doom 3, a Suicide Run usually occurs right after you have taken massive damage and are on the brink of death. With your pitiful amount of health, you decide to sprint through as much of the rest of the level as you can just to see what's there without any regard for enemies so you can be sure to pwn them when you do come to them. Then, when you die, you load your last quicksave.

When you eat dirt for the first time and leave your family to build a shack in the woods

In capture the flag, attempting to assault the enemy base all by yourself, resulting in death.

A run before night fall, usually done with nothing reflective and while wearing dark clothes. Generally done for motivation, usually in populated areas. If not back before nightfall, the runner will be hit by a car, thus the term suicide run.