Some gingers go into dormancy in the winter, others don't. Any idea of the ones you have?
I have never seen/heard of a heliconia that does it. My brother has tried to overwinter heliconias in zone 6 and has never had any luck.

The ginger is Alpinia purpurata. The heliconia is a dwarf Rostrata. Doesn't seem like they should go dormant, but I'm afraid to keep watering them. My GH stays warm and humid all winter. Zingiber zerumbet and Curcuma indora go dormant but Alpinia zerumbet and the Hedychiums do not. Hard to figure these guys out.

No, the alpinias and the rostrata should not be going dormant.
72 degrees should not be a problem for them. Dig one up and check the roots. If they are white and plump- good. If they are browwnish- no good.

I agree that AP and heliconia Rostrata should not go dormant during the winter in their natural environment. But, my experience in growing them in a greenhouse in containers is that they slow down a lot and with regards to the heliconia, may only put out 2 or 3 leaves in the greenhouse from Dec. to March. That's not dormant but is very, very slow growth. Over watering is one of the biggest problems during the winter and I have killed a boat loan of heliconia by doing so. I now water no more than once a week. I killed two pots of Rostrata this winter by over watering and I've been container growing these plants for around 8 years and definitely know better. That's probably your problem. I would stop watering all together until you see new growth even though you might go two or three weeks before seeing new growth.

Steve