According to one of Dr. Sears books (the Birth Book, I think?), the doppler has a bit less energy with regard to ultrasonic waves, but because it is concentrated on one area and for a longer time (especially with fetal monitoring during stress tests or labor), there is still reason to be concerned. I don't know that any study has even been done, like it has with regular ultrasounds, to prove with certainty that it is not harmful.
My personal take is that for checking the heartbeat for a few seconds a few times during the pregnancy, I'm not worried. But, I guess you can never be totally sure, and if it's not necessary, than why do they all use it?

Can't answer that one.