It could be a few things. Assuming it's a gas furnace two things come to mind.
If it's a Pilot type system your fan-limit switch needs to be adjusted or replaced. It's a rotary mechanism that has three set points from lowest temp to highest (fan off, fan on, gas off).
If it's an electronic ignition type furnace via spark ignition or hot surface it could be a few things. It could be a lockout due to failure to ignite, a bad high limit switch or a bad board. There is typically an led indicator on the board that will flash a code. Count the flashes, find the key and know your problem. Typically a lockout will require the thermostat to be turned off and then back to heat to clear, a high limit will require cooling off the house to reset and a bad board just has to be replaced.
Lots of current technicians have no clue how to work on the old pilot systems with the rotary dial limit switch and will try to sell you a new furnace. They haven't been produced in over thirty years now and most techs are younger than that.
If you have any other questions let me know. I've never posted on TexAgs in all my years but felt I could add some value for once.