I have a crawl space under my main entrance of the house. It's about 3' tall. I added a heat duct because the floor at the entrance was always cold. So my question is do I need to vent the crawl space because I added the heat vent. Even if I just cut a hole in the door leading to the crawl space and put a vent cover there. Thanks
Best practice varies with the climate.
If you open up venting between the basement and the dirt crawl, cover the dirt crawl with a moisture barrier.
Consider instead sealing the crawl entrance from the finished basement air, and opening a vent to the outdoors.
In SOME areas you may be allowed to put a fan to the outdoors. Mechanically controlled by humidity, it would offer the best of both cold and warm weather performance.
For research on this topic see,
Moisture Solution Becomes Efficiency Bonanza in Southeastern United States, PDF by Bruce Davis and Cyrus Dastur, AdvancedEnergy.org -Crawl Spaces
Under floor ventilation of wood floors over earth is required by any model code to manage moisture from the earth molding and/or rotting the wood structure. As Justin K indicated, it does make the underfloor area colder, this is why many people block off the vents in the Winter. IMO, if you have adequate floor (and duct) insulation the temperature gained only makes a small difference in energy costs. If the blockage causes moisture issues it will be a false savings.
How serious any moisture problem could be varies greatly by location, climate, soil conditions, and if the earth is covered with a vapor barrier or not. Regardless, you do need vents. Seasonally block the vents if you like, but only if you're sure it will not cause moisture issues.