How bad is it if you inhale a moderate amount of burning plastic fumes? What can happen?

Bad. It depends on how much in distress or how short of breath you are -- you probably should go to a pulmonologist and get a pulmonary function test and at least a chest xray -- you may need to take steroid inhaler to control the inflammatory reaction from breathing that fumes -- some even need to be on a ventilator in ICU for awhile.
Not a good idea. When plastic burns, there are likely a number of chemicals that are released. It is likely that these chemicals are not good to breathe and inhale into your lungs (which could eventually get to the bloodstream). Sometimes, these chemicals can irritate the lungs causing a reaction, such as coughing, wheezing, and difficulty breathing. I would avoid doing so.