Turn phone on with ADB or Fastboot? My power button seems to have busted, is there a way I can boot my phone using ADB or Fastboot or some other command line interface?
When your phone is not turned on it can't have a adb service running (Disregarding the possible technical difficulties it would also be a security issue.)

Try the following instead:
 1. Take out your battery and connect the phone to a charger
 2. Your phone should now turn on.
 3. Insert your battery.

It worked on my current and on my previous phones (ZTE Blade and Huawei U8220) but it may vary by model and manufacturer.