I've recently purchased an Arduino Pro Mini, 3.3V version. I've planned to use it in a diy remote: I've empty a ps2 remote, just left the buttons, and the board will be inside. The remote will be wireless: I use an xbee, which also fit inside the ps2 controller. To power it, I use a one cell Lipo, which will empower the Arduino and the XBee, here comes my issue: As you've seen, there will be many things an a single ps2 controller, so I try to minimize connections and external components. But, the XBee will take more than the maximum 150mA provided by internal regulator of the Arduino. So, either I use an external regulator, or I directly plug the lipo in the VCC pin if the Arduino. Can the board survive? If the voltage is greater than 3.3V, will it damage the board?
The only thing about the 3.3V Pro Mini that is actually 3.3V is the 3.3V regulator. The main chip (pretty much the only other component) is good for anything from 1.8V up to 5.5V.
Running direct from a 3.7V LiPo or Li-Ion will be perfectly fine. You might want to disconnect or otherwise disable the regulator though to reduce current consumption - the latest Minis have a solder jumper SJ1 for doing just that, but older versions you may want to actually cut the 3.3V OUT pin of the regulator or de-solder it entirely.