There are a range of opinions regarding the importance of the information on your used boarding pass when you leave the plane. In general it’s a good idea not to just leave it somewhere; rather, rip it up or shred it after you get back home or to the office.
Enough personal information is on that piece of paper, to either cancel the remaining legs of your trip, or reset the password for your mileage account. http://krebsonsecurity.com/2015/10/whats-in-a-boarding-pass-barcode-a-lot/
But Snopes seems to think it’s not such a big deal. http://www.snopes.com/never-throw-boarding-pass/ .
Either way, it’s probably best to be safe than sloppy.