What could cause the furnace blower to come on constantly and no fire in gas burner?
You have a bad high limit switch, or the furnace is overheated. The high limit switch is in place to make sure the furnace doesn't heat up to the point that it damages itself, or anything around it. If the furnace gets too hot, the limit switch opens. When the switch opens the burners turn off, but the blower continues to run in an attempt to clear the heat. Once the heat is removed, the limit switch should reset (if it's an automatic reset type). In the case of a bad switch, it never resets, so the burners never come on.

Limit switches can go bad with age, but more often than not fail due to overuse. Dirty filters, blocked return ducts, closed registers, really anything that restricts air flow through the system can cause the furnace to overheat. If this happens too often, the limit switch will die prematurely. 

Have a local licensed HVAC technician install a new limit switch.