Does a poor mobile network signal cause faster battery drain?
Quite a simple question actually, does a poor mobile network signal
  cause faster battery drain?


Yes, the problem is three fold:


The phone needs to use more CPU and radio resources to establish a link
A poor signal corresponds to higher packet loss. Whenever loss occurs packets are re-sent. This means more energy is expended to resend the lost packets.
Radios will increase their power to compensate, this means more power will be used.