In the winter should you heat a low/dirt floor crawl space with a furnace vent?

No, its quite the contrary. Heating a crawlspace will prevent mold. But heating a dirt floor crawlspace without taking any steps towards insulating the crawlspace and creating an air seal is going to cost dearly, so there are a few things I would do first to prevent heat loss, air leakage, moisture damage and mold. This page here will have most of the answers you need – 

Best Practices for Insulating Crawlspaces

So here are a few questions that will help us get you more specific answers about how to best proceed – Where are you located? City, state, climate zone, that helps us know what sort of heat load you will have. 
Is the plumbing in between the joists only, or is there equipment in the floor (water heater etc.) that needs to be kept warm? 

Is there currently non membrane at all on the ground, its just dirt?
Is there insulation between the joists? 
Is the dirt floor flat or uneven with stone etc? 
Do you live in a high risk area for Radon Gas? 

There are two main ways to insulate a crawlspace floor, either you can separate it and keep it cool, or treat is as part of the conditioned space and keep it warm. Where your plumbing is may help determine which is best for you, as well as what your budget is for the work that you may perform. To be clear- we don’t perform work so I’m not trying to tap into your budget to fill it, just trying to find the best path forward for you. 
Insulating it would be best, that could be either between the joists, or also insulating the floor. 
 

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