Does being connected to and running on Wi-Fi drain the battery faster then having Wi-Fi turned off ?
Maybe.
The WiFi radio uses less power than the 3g radio. With equal signal strength or better you will save battery on WiFi, provided you set it up correctly. Open the wifi settings and hit the menu key. select advanced, then sleep policy. Set it to never.
OTOH, if your WiFi signal is weak and your 3g signal is strong, you may use more battery running WiFi. Try both ways and see.
On the Droid, using WiFi at home vs. 3G makes my battery last twice as long.