How often should the air filter in the furnace be changed?
Dirty filters not only make the HVAC equipment not last as long, but the low air flow makes them less energy efficient. A dirty furnace does not conduct heat into the air as well and the dirt sticks to the fan blades making it not last as long or spin as easily. 

There are two main types of air filters: 

The 1" thick ones that look like this:

 

And the 6" thick filters look like this:


(source: filterscanada.ca)  

Generally the 1" filters are good for a month or two. The 6" thick ones are good for 6 months. 

There are also 2", 4", and 5" filters available but generally not at the local hardware store. Some folks who have 6" filters use a 2" + 4" filter and replace the 2" one every other month and the 4" once a year. This seems like a hassle to me, but it might save you $50 a year if you have 6" filters.