Do ANR headsets affect pilots' perception of engine problems?
Although its possible to detect engine problems audibly, I don't think that's the primary mechanism (except for engine out) by which pilots are expected to detect engine anomalies. Part 91.205 requires that every aircraft be equipped with the following instrumentation, and these should be the primary mechanism by which engine malfunctions are detected. 


  (4) Tachometer for each engine.
  
  (5) Oil pressure gauge for each engine using pressure system.
  
  (6) Temperature gauge for each liquid-cooled engine.
  
  (7) Oil temperature gauge for each air-cooled engine.
  
  (8) Manifold pressure gauge for each altitude engine.
  
  (9) Fuel gauge indicating the quantity of fuel in each tank.


That being said, ANR headsets attenuate sound, but don't eliminate it entirely. Having flown with an ANR headset for most of my flying career, I can certainly hear variations in engine noise in the mag check and prop exercise during the runup, as well as during throttle and prop changes. I can also hear a fouled plug during runup, as well as verifying it visually on the gauges.