Faraday cage with a RF WIFI router inside - does the electricity "evaporate"?
No, radio waves are not "electricity" in that sense. Electromagnetic waves are produced by electric charges moving inside a conductor (the antenna), but the charges never actually leave the conductor and go somewhere else.

The RF power emitted by the router reflects from the inner surface of the cage, and bounces around until it encounters something that can absorb it, such as the plastic case of the router and other nonmetallic items inside the cage. These items will get slightly warmer as a result.

The router itself consumes a significant amount of electrical power while operating. A small fraction of this power gets emitted as radio waves, but the vast majority of it gets turned directly into heat within the components of the router.