In Destiny, I got invited to a fireteam. How do I leave the fireteam without having to log out? Is it different for different systems?
It depends on where you're leaving from.
At the Tower
There is a button on your controller that you can press to bring up the Return/Go to Orbit and Summon Vehicle options. (On PS4, it's the giant touchpad you can press, and on the PS3, it's the Select button. I'd assume it's the Back button on the Xbox 360, and the button to the right of the left joystick on the Xbox One)
Once you bring it up, there's an option that'll say "Leave Fireteam".
You should be able to just press whatever the cancel button is on whichever console you're playing on (B or Circle).
I think I'm missing some more situations, but these are the two that I generally leave fireteams from.
If you're the fireteam leader, first promote someone else to leader by selecting their name in the fireteam list (the roster) and selecting promote. Then just go to orbit and it will drop you out of the fireteam.
(only tested on xbone)