If you want anything close to an accurate answer. You're going to have to tell us a bit more about the wiring between the thermostat and the furnace, or include some photos of the wiring. And include make, and model numbers for all the equipment involved.
Most furnaces manage the blower themselves during a heat call, i.e. the thermostat simply energizes the W wire. Some also manage it during a cool call, but not all do. So when calling for cooling, the thermostat usually energizes both Y for cooling, and G to turn on the blower.
Since you say the blower doesn't come on for cool calls, or when the fan switch is moved to the ON position. That points to a problem with the "fan circuit". Between the thermostat and the furnace, there should be a wire (usually green) connected to the G terminals of both devices. When this wire is energized, it tells the furnace to start the blower. Some furnaces have an on/off delay, so the blower may not come on immediately. On some units, this delay is adjustable.
Since you haven't told us the make and model of any of the equipment, there's no way to be specific about what happens in your furnace. Some furnace control boards have terminals for HEAT, COOL, and FAN. While others may only have HEAT, and/or FAN. The motor speed wires are connected to these terminals, to allow the furnace to run the blower in different speeds depending on the situation.
The first thing I would do, is test for voltage between the grounded ("neutral") and the COOL terminal on the board, during a cool call. If you measure about 120 volts (line voltage), that means the board is energizing the terminal to turn on the blower. This means there's a problem between the board and the motor, or with the motor itself. If the terminal is not energized, then you'll have to start moving "back" through the circuit.
If the furnace is signalling for the blower to turn on, but the blower is not starting. You could try connecting a different speed wire to the terminal. Since the blower works for heat, you might try putting the wire that's on the HEAT terminal on the COOL terminal. If the blower fires up, you know it's a problem with the motor. WARNING: Don't run the system this way for long, you're just checking to see if the blower starts.
If the furnace is not signalling for the blower to turn on (no power to COOL terminal), then you'll want to start by making sure the thermostat call is reaching the furnace. Disconnect the wire connected to the Y terminal, and set the thermostat so it's calling for cool. Measure voltage between C (or the common side of the transformer), and the wire that was connected to Y. You should get about 24 volts AC. If you don't there's a problem with the thermostat, or the wire between the thermostat and furnace. If you do, then the problem likely lies within the control board.
Without more details about the system, it's impossible to offer any more specific advice. It's possible that there's an open limit, which is preventing the blower from turning on. However, without a schematic, there's no way to know if there are any limits that might be involved.
Usually A/C, heat, and fan run the blower at different speeds. It's possible that the speed the A/C uses is dead.
Any easy test, is to swap the motor speed wires connected to HEAT and COOL on the board. Then see if the blower starts with the A/C. If it does, that means the speed used by A/C is dead. Depending on the system, you might have to replace the blower, or simply run it at a different speed for A/C.
WARNING: Don't run the A/C with the wrong blower speed for long, as it could damage the system.
If it's an ECM motor, there may not be a way to select/change the speed manually. Or the speed selection might be adjustable using dip switches, or jumpers.
Best Answer
Bad Blower Motor Coil
Furnaces typically have multispeed blower motors, and use different speeds for different modes. If the coil for the speed the heat uses is dead, that would cause the problem you're seeing.
You can use an ohmmeter (or multimeter set to measure resistance) to check if the motor coils are good. You'll have to measure the resistance of each coil (while isolated from the rest of the system), and compare the results to the manufacturer's listed values.
Bad Control Board
There are a few things that could go wrong on the control board, that could lead to the issue you're seeing. The first thing to do, is to determine if power is ever getting sent to the motor at all.
Set your multimeter to measure AC voltage, and connect the common lead to the neutral terminal on the system. Hold the door lockout switch in, so that the system will try to fire. Once the burners ignite, listen closely to hear any relays closing.
Most furnaces use timers to delay the blower, to allow the heat exchanger to heat up. Once the preset delay has elapsed, a contact should close and send power to the HEAT terminal on the control board.
Put the other lead of your multimeter on the HEAT terminal, and see if you get a voltage reading. If you do, then the problem is with the motor or the wiring to the motor. Check all connections, to make sure they are clean and secure. If you don't read voltage, you'll have to either troubleshoot the control board, or simply replace it.