I am almost positive that this game doesn't auto backup to anywhere online, but if I happen to wipe or switch phones, can I perform a manual backup of my game state prior? If so, how ? I am playing this on an iPhone 4s running 5.0.1.
The game actually does backup - it backs up to Game Center.
From the app description - "Game Center integration to let you see your friends towers, back up your game and earn awards."
I have used the Game Center backup myself - I deleted the app for a while when I got bored of playing it, and then I reinstalled it after a couple of months. When you restart the game and sign in to Game Center, a message will pop up alerting you to a saved copy of the game (if there is one) that you can choose to use instead of starting over.
You can use iCloud/iTunes as well to backup your game. A post on the GetSatisfaction forum from Dani Rudolph, a Nimblebit employee, suggests following this Apple guide that helps you through the backing up and restoring process.
If you use iCloud/iTunes to back up, you can then use this Apple guide to transferring the information from one iPhone to another.
You can back up your files with this Whatsapp Recovery for iPhone that will restore your deleted files you accidentally erase.
Tiny Tower backs up your game to Game Center but does not sync between devices unless you're starting fresh with no data on your device. Compared to NimbleBit's Pocket Planes, which does have iCloud support and lets you play between devices, Tiny Tower's backup is only useful if you lose your save or get a new device.
(NimbleBit, if you're reading this, please add iCloud support to Tiny Tower.)