Can I move a receiver from the bedroom to the living room?

If it's a wireless receiver you can move it anywhere you like so long as the Wireless Access Point connected to your gateway can "see" it (i.e., the distance and/or the construction material the signal passes through are not a problem).  The only real way to know that is to try.  You may or may not have to re-pair the box with the AP.

If it's a wired (not dvr) receiver you can still move it of course.  It's your problem how you string the cabling. The ethernet cables just plug into the ports in the back of the gateway.  Obviously wireless boxes are way more convenient.