Does my furnace suck air from outside?
I think some of the answers here are confusing because they don't really address that your furnace has two different air intake/outlets.
One is for the combustion system. Air is pulled in from somewhere (usually outside, probably the metal duct on the right), mixed with natural gas, and burned. All of the outside air and combustion products go out of the chimney to somewhere definitely outside. Very hot air from the burning of gas heats a metal piece called the heat exchanger.
The &quot;makeup air&quot; mentioned elsewhere would be if the combustion loop sucks air from inside the house instead of through a duct from the outside. In this case some air from the house is blown out the chimney and is replaced by air seeping in through the cracks. This would not be code compliant where I live, for example.
The other air loop is through the ducts in the house. Typically a furnace will suck air from inside of the house (probably the vent behind your couch), pass it over the heat exchanger, and blow the now hot air out of the vents in your rooms. This typically does not pull a significant amount of air from outside because that would waste a lot of $$$ by displacing heated air to outside. Houses are not airtight though.
Your furnace is obviously gas because of the supply pipe on the lower right, and the presence of a chimney on the top of the unit.
There is really no way for anyone here to tell you whether it is safe or not. Age and presence of recalls are good places to start though. Adding a CO detector and making sure you have enough smoke detectors would help significantly.

We live with someone with COPD and I also have asthma,

If this is a matter of LIFE AND DEATH, there is not enough information in your post to provide an absolute answer. YOUR SYSTEM MAY NOT BE NORMAL.
Have you ever changed the furnace air filter?