Sort of important question

Um...

I would say no you wouldn't lose your hair...
It's no different from wearing a hat, helmet or a hoodie. Last I checked those aren't proven causes of hair loss. >.>

 

There is no correlation between hair lose and headset usage. your mom may be thinking about cancer. I did research and found this.
Wired Headsets and Radiation
A wired headset is an earpiece-microphone device that fits on your ear, and connects to your phone via a wired cable. These do produce radiation, as does any wire that carries an electric current. However, the radiation from a wired headset results in weak radio waves at audio frequencies -- difficult to detect, even with sensitive equipment, and several orders of magnitude weaker than the radio signal the cell phone produces. Using a wired headset is one of the safest solutions available for cell phone users.

 

Eh...parents will say anything to make you paranoid :-/

 

Look at your question in this way. Using a wired headset is sort of like using a cell phone. Your phone gives off radiation, but it's so little that it has no effect on the human body. The same goes for a wired headset. Being around parents with stupid ideas is probably less healthy for you.

 

silly parents always trying to trick us >_>