Furnace fan will not turn on when in "Auto" mode. If switch to "Manual" mode on thernostat, fan will come on, but doesn't blow warm air.

Blower is running, check thermostat, sounds like it is setting on fan mode. Make sure thermostat is turned to heat not accidentally on cool and fan switch on thermostat in in auto mode not fan mode.

The fan will not blow until the furnace lights when in auto mode. You may have a defective igniter or gas valve.

Sounds to me like it is going off on a weak high limit switch. With the fan on it will prolong the event but will not stop it. I would check all of the high limit switches. It would most likely be one close to the heat exchanger.

Go to your thermostat and access the fan mode. Change the fan mode to Automatic, AU. This will stop the AC fan from blowing when the furnace is turned on. If the fan mode is on Hi or Lo and the furnace is turned on it will turn on the AC blower. Bad design...

check for loose wiring to Tstat, or the new stat has failed.
if the stat has a fan on switch and when on the fan works then replace the stat

Is your tstat fan selector in auto?? Does the fan ever shut off?

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Hi #1 check thermostat : see if the FAN ON/AUTO switch is set to( auto ) and not( on)...If this is not the case turn the furnace switch off for 20 sec. and back on...did it stop? IF so the blower relay is sticking and must be replaced.
AUTO=BLOWER ON DEMAND/CYCLING
ON=BLOWER 24/7 all the time..
hope this helps

Check your thermostat to see what your fan switch is set on. There should be two settings - auto and on. If it is set on auto, the fan only comes on when furnace runs. If the switch is set on on, the fan on your furnace is running all the time. Which means when temperature in the house reach what you have set on your thermostat, the furnace kicks off, but the fan keeps running and blows cool air. Some people run their fan all the time, especially if they have allergies, because that's a good way to clean the air.

Depending on the model of the furnace, the blower is either a 2 or 3 speed. When the furnace operates in heating mode with the thermostat in the auto position, the furnace starts the fan when the furnace reaches a specific temperature or after a set time delay, but it operates the fan in low or med low speed on heat mode, when you put the thermostat fan switch in the "ON" position, the furnace operates the fan in "HI" speed, the same speed used for cooling mode. You may have a blower moter that is starting to short. I would recommend having a professional replace the motor before now before it becomes an emergency in the middle of the night during a cold spell.

It sounds like you T-Stat is messed up. Look in your manual to see if there is set up instructions. If not I would change the T-Stat.