Actually it depends on the serial port. The old style 1489/1488 IIRC, were able to source quite a bit of current, and ran at +/- 12V (ish). A lot of the newer stuff has MAX232 or equivalent chips, and may only produce +/- 3 to 5 V. I've powered things from serial ports before, OR the control lines (DTR, RTS) and the TXD line together using diodes.
You are unlikely to damage anything, since most RS232 interfaces are current limited, but don't blame me if you do.
Found this pdf, knew I had it somewhere!