The draft pressure switch on a gas furnace is a safety device designed to shut down the furnace if it's not receiving the airflow it needs for combustion. It does this by interrupting the electrical current that ignites the furnace. Your pressure switch might fail to close for one of several reasons. Before you investigate them, you should make sure the switch itself isn't faulty.
Use a volt ohm meter to test your furnace's pressure switch after you turn off the power to the furnace. Before you begin, touch the tips of the probes together to zero them out. Then set the meter to 24 volts. Attach the black probe to a metal part of the furnace to ground it, then place the red probe on the metal tube that connects the pressure switch to the draft inducer motor. Check the reading. If it's less than 24 volts, there's something wrong with the switch. A reading between 24 and 28 volts indicates that the problem is not with the switch.
The pressure switch might fail to close if a vent is restricted. Pull the hose off the pressure switch and test it for clogs by blowing air through it or by listening for airflow when the draft inducer turns on. If you can't do either one, the inducer nipple is clogged. Clear it out with a drill bit or a piece of coat hanger. A stopped up drain line on a condensing furnace can cause the same problem. Use a wet vacuum to reopen the drain line.
The combustion air blower, which is also known as an inducer blower, provides air to the burners. It also clears byproducts from the combustion chamber. The combustion air blower starts blowing air when the thermostat calls for heat. The purpose is to create a draft. When there's enough draft, the pressure switch closes because of the negative pressure created by the combustion air blower. This lets the control board know it can fire up the igniter and then open the gas valve. However, if there's a problem with the combustion air blower, the pressure switch will fail to close.
Pressure switches can also fail to close if carbon deposits are on the surface of the heat exchanger, although this is usually more of a problem with propane gas than natural gas. High winds can also be responsible for a switch that stays on.