Here is some info from an officer with the IDNR Police:

Almost a simple answer----, we do not have a viable breeding population of mountain lions or cougars (same thing) in Illinois--have not for a long time. Reread that sentence, I did not say you won't find them in Illinois--I said there is not a viable breeding population. That said, we have had 3 documented mountain lions killed in Illinois in the last 10 years--I believe 2 were south american species (speculated that someone released them) and one originated from a population in the Black Hills of South Dakota (a young male--likely traveled here on its own, they are known to range far.)

Cougars are not protected in Illinois under the Wildlife Code.

HOWEVER, there is the Western mountain lion and the Eastern mountain lion (sorry I don't know the Latin species names). The Western obviously occurs out west in strong numbers and is hunted and managed in the western states. The Eastern is only found in southern Florida from a very, very limited population. The Eastern mountain lion is Federally protected as an endangered/threatened species. Therefore the Eastern mountain lion is protected in Illinois as well as Federally under the Endangered/Threatened species act.
Now that I've said all of that, the odds of you seeing a mountain lion are very, very small. I've investigated numerous complaints and spend a large amount of time outdoors and have never seen one or sign/evidence of one. It is even far more unlikely that it would be the endangered/threatened Eastern mountain lion. It would be much more likely to be a Western or South American species that would not be protected in Illinois.
So, to answer your question, yes you could shoot a mountain lion in Illinois if it is not an Eastern mountain lion. If you did shoot one, immediately call the local game warden or biologist and they can do the appropriate studying and testing to determine the species and, (every bit as important), possibly investigate other clues as to how it got there. Will you get in trouble? No, not if you have the permission of the landowner/tenant and it does not turn out to be an Eastern, (the chances of that are very low as I mentioned above.)
If you have any other detailed questions regarding mountain lions, you can try to contact Dr. Clay Neilson at SIU Carbondale, or get a copy of the June 2006 Missouri Conservationist which has an extremely good article about mountain lions. I hope that helps.