The only way to "dodge" a Tower shot after it has been fired is to become untargetable. Several champions have this capability, but every champion can become untargetable through the use of Zhonya's Hourglass. However, to survive a tower with Zhonya's, one of several things must happen:

 1. *Get out of the tower's natural range before using Zhonya's* - The thing about Zhonya's is that it freezes you in place. It does you very little good if you use it right under the tower as the tower will just shoot at you again as soon as you unfreeze. Get out of range with a dash, Flash, or any other movement ability and pop Zhonya's.
 2. *Activate Zhonya's with nearby minions* - Zhonya's will remove you from the targeting list, essentially resetting the tower's focus. If done while in the middle of a minion wave or right before one arrives, you can get the tower to focus back on a minion while you unfreeze and stroll to safety.
 3. *Stall for time* - If you have an allied champion with an emergency save ability (Soraka, Zilean, Shen, etc.), you can stall for time with Zhonya's and then try to get that champion's attention before the effect wears off. This can be especially useful if the champion is just waiting on a cooldown to come back up before they can save.

Be aware that if you damage an enemy champion as you come out of the Zhonya's Hourglass effect, the tower *will* target you again (unless you're out of range due to option #1). Be careful when tower diving as a champion with damage-over-time abilities like Malzahar.

While I've mostly talked about Zhonya's with this answer, this applies to *all* abilities that render you untargetable (Fizz, Vladimir, Bard, etc.)