If I am pushing my aircraft into the hangar with a tow bar, at night, with the engine off, am I required to activate my rotating beacon?
The AIM gives good advice, but that is really all it is. It is acceptable methods of complying with the regulations, but you are not required to comply with what the AIM says. In questions of what you are required to do, especially in questions of whether you might be at risk for a violation, we want to look at the regulations in 14 CFR.

14 CFR 91.209 states, in pertinent part:


  No person may:
  
  (a) During the period from sunset to sunrise [...]—
  
  (1) Operate an aircraft unless it has lighted position lights;
  
  (2) Park or move an aircraft in, or in dangerous proximity to, a night flight operations area of an airport unless the aircraft—
  
  (i) Is clearly illuminated;
  
  (ii) Has lighted position lights; or
  
  (iii) is in an area that is marked by obstruction lights;
  
  (b) Operate an aircraft that is equipped with an anticollision light system, unless it has lighted anticollision lights. However, the anticollision lights need not be lighted when the pilot-in-command determines that, because of operating conditions, it would be in the interest of safety to turn the lights off.


Note:
Position Lights are commonly called Nav Lights
Anticollision Lights can include what are more commonly called Strobe Lights and Beacon Lights (See AIM 4-3-23)

Lets first look at what the above regulation says about operating requirements. Is your given scenario (engine off, etc.) an example of operating the aircraft? 14 CFR 1.1 gives the following definition of operate:


  Operate, with respect to aircraft, means use, cause to use or authorize to use aircraft, for the purpose (except as provided in §91.13 of this chapter) of air navigation including the piloting of aircraft, with or without the right of legal control (as owner, lessee, or otherwise).


By that definition I would argue that towing, pushing, or otherwise moving an unoccupied aircraft with engine(s) shut off would not be operating that aircraft anymore than leaving the aircraft parked overnight would be operating it, since you are not using the aircraft for the purpose of air navigation. Thus I would say that you are not required to follow any rules that pertain to operating the aircraft, such as that any installed anticollision light be lighted.

(This is in contrast to an airliner pushing back with a load of passengers, flight crewmembers at their stations, electrical power on, APU running, etc., which would be operating, since the aircraft is being used.)

This leaves us with the Park or move paragraph. This paragraph specifically pertains to "in, or in dangerous proximity to, a night flight operations area of an airport".

What is a night flight operations area? Absent any proffered definition, the plain reading of the phrase—as I read it—would be an area used, or subject to use, for flight operations at night. In other words, any public use airport with an apron or taxiway that is open to flight operations at night could and should be considered—by my reading—a night flight operations area. Whether the area is a movement or non-movement area shouldn't matter. Could someone else reasonably taxi past you (or into you!) at night? That's probably a night flight operations area.

Thus, for your given scenario, if you are pushing your aircraft back into a tie-down spot on an apron that is open to night flight operations, by regulation it must be either:

a) clearly illuminated;
b) have lighted position lights; or
c) be in an area that is marked by obstruction lights.

I typically see airport ramps clearly illuminated by large overhead lights. This clearly meets the first requirement. Another means of compliance would be to turn on an internal hanger light to clearly illuminate the airplane when you push an airplane in off a taxiway into a hanger.

You are also good to go if you are in an area that is marked by obstruction lights. I don't think I have ever seen these demarcating an area where you would be pushing or towing an airplane (I'm thinking of those lighted construction barriers), but if you saw them I suppose you would know.

In the absence of clear illumination or an area marked by obstruction lights, if you are going to park or move the airplane at night, you had better turn on the position lights. And this, I think, is where you could be at risk for a violation.

Note that, even in that case, the requirement is for position lights, not the beacon. Unless the aircraft is being operated, there is no requirement that any installed anticollision light be lighted.