Edit: To be clear, I mean keeping long term and unintended to be keepable by Rockstar.
An example is this:
Some missions lower the friction of train tracks for non-trains for their entire duration.
If you load a save or start a new game, then the friction stays.
Another example is taxi warps in VC - they also stay when loading a safe so you can warp by saving, loading some old savegame that has a mission near your destination, starting that mission, then loading your new savegame and dying. That is not too helpful, though. ^^
I'm interested in everything, even the most useless stuff.
Edit: Another example is that failing Blood Bowl due to the timer running out lets you keep the one free death - you keep weapons [if you have a parachute as you die then it will be "broken" and you need to save&load to fix it].
I am also interested if the effect can only be kept with glitches. For example, if you start Taxi Driver while Mountain Cloud Boys is running you keep the "ignored by cops" effect until mission_cleanup (which runs at the end of every mission). Some effects might be keepable for longer, though.
Applications are also interesting. I am mainly interested in speedrunning strategies but anything is welcome.