Question: I just moved into a new home that has a backyard composter (about 80 gallons). The composter has been sitting unused for an unknown amount of time (at least 3 months).

How can I assess the health and status of the compost pile, and then what should I do to revive it?

Answer: 
I agree it is a good idea not to use the compost left in the bin on anything edible or valuable. Just scrape it out and start again. They might have been composting meat, in which case it will be full of maggots and harmful bacteria. The idea is, if it smells spread it out in an out of the way place and let the air and the worms get to it.
If you're concerned your compost pile is no longer biologically active, restart it.
There's bacterial start up's that can easily be purchased. Another way to speed it up is to Heat it up. Fresh urine (98.6 degree's) can activate a pile too, another strategy is to use hot water; because temperature drives thermophilic activity.Dig a donut shaped hole in your pile Boil some water and soak a big piece of crumbled up cardboard, hen immediately add the cardboard to the bottom and press it like a bowl and dump the hot water then back fill the hole and add the bacteria. Heat rises so a warm environment is perfect for them. Also you've mixed and there fore aerated the pile