As the title says, is there any way to run faster than the turret attack (the missile like thing) so that you can arrive spawn before the attack reaches you, and therefore can't kill you? You know, when you have low health, and you are targeted by the turret? Is there any item sets or things like that?
Once you're targeted and the turret fires a shot, there is no avoiding the shot. It is locked in, unblockable*, and unavoidable*.
Now looking at the second part of your question; yes, you could probably heal enough to avoid a mortal blow. If you were teleporting back and the turret fired a shot as you poofed, you would heal enough by the time it crossed the map. If you were to outrun it, you would need a team set up to assist you as well as using multiple shurelyas for ms boosts.
Runners: Sion (ulted), Hecarim (his e), Teemo (w passive/active), Rammus (powerball), or Rek'Sai (ult to a tunnel in well? This might be pushing it).
Assisters: Janna (passive), Orianna (w ability), Jayce (acceleration gate), Galio (whatever the heck his e is) and/ or Zilean (his e ability).
So in all, yes it is possible. But it is highly impractical unless you want to test it out with some friends just for kicks.
*- Pantheon's passive can block a turret shot, Fizz can use playful trickster, Elise can Rappel, Vladamir can pool, zhonyas can be used, etc. Basically any spell that makes you untargetable will cancel a turret attack.
You can do this with a speed steroided Hecarim to effectively run twice as fast as the turret shot and have more time to regen and possibly get to that Soraka desperately running to you to give heals. However, the items needed to get this speed are generally not viable, but can still be fun to do if you're messing around in blind.
As a Kata Main, I've experienced this issue quite a bit. When i towerdive and get a kill and then Shunpo away, if i am able to get sufficiently away from the tower, the tower shot damage is dealt, irregardless of the visual animation of the shot hitting you or not. However, once the damage is dealt, the animation becomes a marshmallow, i.e., no damage is dealt once it hits you.