Here is my short story:
I used to litter cigarette butts all over the place like it was my mission in life. Then, one day I moved to Yosemite National Park. While I walked to work, I littered a butt on the bike path. I noticed that same very butt on the way back home and thought to myself 'wow, that's weird.'
So the next day comes and sure enough, on the way back to work, there is the butt again! A slight tinge of guilt hit me, but I still left it there. Well, 2 more days of stubbornness went by and finally, I leaned over, and picked it up.
Ever since that experience, I stopped littering cigarette butts (the only thing I have ever intentionally thrown on the ground). Living in a city, town, whatever, we kind of grow reliant on street sweepers, city employees, and other property owners making sure the area looks nice.
Now I know the Disc Golf Course is no national park, but WE are its 'employees.' These courses are usually tucked back in areas that are somewhat hidden from the public eye, and its up to the users of that area to take care of it. That would be us. Do you ever look at those huge piles of cigarette butts? Guess what, A BUNCH OF THEM ARE YOURS!!! If you expect things to get cleaned up for you, go play tennis, or something on a court that gets hosed down each day. Part of enjoying a nature based activity is respecting nature, and if you can't do that then its probably best to pick up another sport.
Oh, and several people have given me that bullsh-t response that 'cigarette butts decompose.' So do you, over time, but in the meantime, its just an ugly stinky reminder of your irresponsibility.