I recently moved into a house that belonged to my sister and her family. After doing some checking around, I noticed she didn't have a filter in the AC unit. Apparently it has been running for months with no filter. Fantastic! Now that it's cold weather, am I at danger if I turn the heat on? It's not a gas furnace but even with all of the dust and even her dogs hair, can it catch fire? Suggestions? I don't want to put my family in danger.

Investing in cleaning and inspecting your duct work, and heating/air conditioning units as you replace that missing filter should serve you well.

If it is a heat pump, very little possibility of fire. If it is a resistance, all electric, much greater. Have the system inspected and cleaned if necessary.

The filter accomplishes two things: It keeps the dust level down in the home and it keeps the dust off the furnace components. Since you have a heat pump it will allow dust and apparently hair to accumulate on the coil. This will lower the efficiency. Most likely not a fire hazard. Have the coil cleaned or do it yourself.