Will a V.I.P wait in the elevator in Tiny-Tower forever - Or until I make use of him/her?
Solution 1:
For the most part, they will wait there until you want to make use of them. However, if you tap on them to start the elevator, you will be forced to deliver them to some floor before you can take any other actions such as restocking or constructing a new floor. Be aware, however, that as long as the VIP is waiting at the elevator, you will not have any other regular elevator customers.
The only occasion that I've seen where I've "lost" a VIP was when I went to view a friend's tower. Otherwise, they have always waited in the lobby until I decided to make use of them.
Solution 2:
The vip status is saved in the iOS memory so you will only lose your VIP if you access your friend towers or close down the game from iOS.
Additionally, any operation with your device that would flush the RAM, such as rebooting the device or crashing, will wipe your VIP.
EDIT: Wanted to mention this can all be tested with normal elevator patrons.
Solution 3:
If you have version 2.0 you will be able to save V.I.Ps but if you want to use him or her you can press 'use now'. I think you can save 5. If you don't have this new, version you will be forced to use them so you cannot stock, buy or do anything apart from use the V.I.P. Also no regular customers will come if the VIP is in the elevator. If you view a friend's tower your VIP will be lost.