In summary, based on FAA/NTSB information it's possible in theory but no one really knows how individual headsets behave and there's no record of ANR headsets contributing to an accident.
First, according to a very short FAA information paper on Noise Attenuation Properties of Noise-Canceling Headsets, it's possible that they can mask important sounds:
While this technology can have many beneficial effects such as
providing clearer communications, reduced pilot fatigue, and added
comfort, electronic attenuation of important environmental sounds and
alarms may occur
But the paper has no specific information or examples, and its only purpose seems to be to encourage operators to do their own testing (which is reasonable, since headsets can be very different):
Evaluations should be conducted while both on the ground and in flight
during normal operating conditions to ascertain if any audible alarms
or other environmental sounds, or combinations thereof, can be
detected while electronic noise attenuation is on and active.
That paper is from 2007 and a letter of interpretation from the same year on headset requirements in part 121 operations implies that the FAA's main concern with ANR headsets is (or was) that the batteries run out and that might cause problems:
The FAA is particularly concerned, however, that Active Noise
Reduction (ANR) headsets and headset adapters [...] that rely on
battery power are subject to failure when [...] batteries discharge
under normal use
And the FAA's general brochure on Hearing and Noise in Aviation mentions ANR headsets without any warnings or recommendations.
Finally, the NTSB's accident database has only one specific reference to an ANR headset (that I could find) and it explicitly ruled out any effect:
On January 5, 1998, the NTSB IIC used a Cessna 208B to determine if
the Telex model ANR-4100 active noise reduction headset worn by the
pilot at the time of the accident could have eliminated his ability to
hear stall or fuel selector warning horns. All aural warnings were
clearly audible with the engine operating and the headset active.
The information I found online on ANR headsets is overwhelmingly positive and I couldn't find any informal reports of pilots mishearing engine sounds either.