Can we capture the hot, humid air from a gas-powered clothes dryer during winter?
The dryer vent and the combustion vent are one and the same.  If you try to use the dryer vent for heating you will have two issues, first excess moisture and second carbon monoxide.  Gas dryers get their efficiency by directly venting the combustion into the damp clothes which then by the way of evaporation drastically reduces the temp of the heat.  It would be near impossible to modify a dryer to separate the combustion from drying.

Then there is the idea of using a heat exchanger to extract the excess heat.  But I don't recommend it since there really isn't enough excess heat in the line to keep the temp above 100C/212F at the vent exit (below this you risk condensation).  If you get condensation on a cold day, you then risk ice, which then risks blocking the vent, which then risks carbon monoxide poisoning.

Also take a look at the fact there isn't any commercial products to do this.  This speaks to how unfeasible it is to use this waste heat.