Just found out that my wife changed the last furnace filter and inserted it backwards. This was noticed after the ac was not cooling the house very well. I went outside and noticed that the coil itself was frozen over. I shut it down and waiting for it to unfreeze currenlty. My main worry is that this could cause permanent damage. Thoughts on this? Is it probable that this would cause permanent damage?
If liquid refrigerant gets in the compressor, it can indeed cause damage. The compressor is designed to compress gas, trying to compress liquid can burn out the motor.
Let the system defrost. Fix the filter. Then turn it back on. If the system still isn't cooling, you'll have to troubleshoot it yourself, or contact a professional to do it. Make sure you give the system plenty of time to defrost. If the condensing unit was iced up, the evaporator is likely a solid block of ice.
Don't worry too much yet, everything could be just fine. Burning out a compressor isn't very common. You'll likely be okay, as long as you didn't run the system like this for a long time. Which I'm guessing you didn't since it won't work well, and you'd probably notice it wasn't working fairly quickly.
The backwards filter may not be the only problem, but may have been just enough to expose other problems.
Make sure all the registers and returns are open, and free from obstructions.
Closed/obstructed registers and returns, cause lower air flow through the system.
Check the evaporator coil (or have a pro do it), to make sure it's clean.
A dirty coil reduces the air flow, and reduces the transfer of heat.
Check the blower motor (or have a pro do it), to make sure it's running at the proper speed.
A slower motor moves less air over the coils, which reduces the amount of heat transferred to the refrigerant.
Check (or have a pro check) the refrigerant level.
A system that doesn't have the proper amount of refrigerant, will not function properly.
The backwards filter caused reduced air flow through the evaporator, which reduced the amount of heat transferred to the refrigerant. Because of this, the refrigerant was likely not boiling off as soon as it should have. This caused the coils at the beginning of the evaporator to be very cold, which caused the normal condensation that is on the coils to begin to freeze. As the coils freeze up, the air flow is reduced even more. As the air flow reduces, less heat is transferred to the refrigerant. This causes a snowball effect, which carries on down the line.
Eventually the entire evaporator coil is in a solid block of ice, and the suction line (larger diameter pipe) is also frozen. At which point, liquid refrigerant will be making its way all the way back to the compressor. The compressor will now be trying to compress a gas liquid mixture. Since it cannot compress the liquid, the compression rate will drop. This causes the temperature of the compressed gas to be lower than it should be. At the same time the liquid in the mix will start to boil in the condensing coil, which will drop the temperature of the coils. As the condensing coil temperature drops below freezing, any moisture in the air around the coils will start to freeze.
If the system is allowed to run like this, the refrigerant in the compressor will be more liquid than gas. Since the liquid cannot be compressed, the compressor will struggle to operate. Eventually, the overworked compressor will give up.