Does anaerobic exercise stop or significantly inhibit fat burning, and
if so, for how long? I would like some links to scientic research that
discuss this phenomenon.

Only during the act of muscle contraction -- assuming the "anaerobic"
exercise is utilizing glycolysis and/or the one enzyme systems to
generate energy.  During recovery (an "aerobic" process) fat oxidation
will resume.  Fat oxidation can be quite high after strength training
assuming carbohydrate beverages are not consumed during or after the
workout (Acute EPOC response in women to circuit training and
treadmill exercise of matched oxygen consumption.
Eur J Appl Physiol. 2005 Aug;94(5-6):500-4).  While RER is a poor
indicator of substrate use (it is a whole-body measure with some CO2
produced by non-metabolic factors -- primarily buffering of H+ ions
derived from the lactate dehydrogenase reaction), it was elevated
during the "anaerobic" exercise, which is indicative of carbohydrates
being the primary fuel during the actual workout.  Post-exercise
however, the "anaerobic" trial lead a great suppression of RER
(indicating more fat being oxidized).