Do transmission lines induce electric or magnetic fields in towers?
Do these fields affect transmission towers?


To some extent.


You have electric field simply because of the voltage on the wire. This static electricity has an effect on nearby objects. Air ionization, attraction forces, etc. 
The magnetic field induces currents (and voltages) in conductive structures. BUT, the effect on towers will be minimal simply because power lines and comms towers are perpendicular to each other. So pretty much no inductive interaction comes into play.


What's more, such structures are always grounded. So there's no voltage on them. The only thing you may be able to detect on the tower is its magnetic field originating from the 50Hz current flowing up and down the structure. However I said this current will be minimal, so yeah.

If you'd be able to measure currents in power lines from any of that, I have no idea.