Does old boarding pass contain enough information that it should be considered sensitive and dealt accordingly? (shredding the printout instead of throwing it to recycle bin) For example: can someone misuse my frequent flyer number?
If the boarding pass is for an itinerary which is not yet complete (e.g. it was the outbound leg of a round trip, and you haven't yet taken the return), then there is a risk. The boarding pass usually contains your name and confirmation number, and this is often sufficient information for someone to change or cancel your remaining flights. I don't think they could easily profit from this, but they could at least inconvenience you. (And if the airline charges a fee to change the flight and/or change it back, or the ticket is non-refundable, you may have some trouble convincing them that you didn't authorize the change and they should put it back without charge.)
Once the flight has passed, the boarding card holds no information of value. However, it does contain your name and may also contain your frequent flier number - for someone mining data on you, this is valuable information.
Also keep in mind that the boarding pass stub is one of the few pieces things that serve as (legal) proof that you were onboard the flight.
This may come in handy; should there be a dispute of any sort (for example, mileage redemption, etc.)
It is really not that useful to law enforcement - as they can request the flight manifest directly from the airline/DHS.