I have an old Dynavent DNV40 natural gas furnace that when I turn the thermostat to heat, the fan will kick on but it never tries to ignite the gas. On rare occasions, it will ignite the gas, but after about 10 minutes of operating, the gas cuts off and the fan continues to blow. I replaced the old mercury Honeywell thermostat thinking that might be the problem, however the new one is not doing any better... At least I have gotten rid of the mercury, I guess... Any solutions??

Sorry but I never heard of that brand. Are you in the USA?

I am in the US. It is made by Suburban Manufacturing Company (Dayton, Tennessee).

Call them

Suburban Manufacturing (http://www.suburbanmanufacturing.com/)

If this is the window unit there is an internal heat sensor. I recommend u get a tech. They are not easy to repair.
This is not a thermostat problem.