From what I've gathered, Foursquare allows you to toggle a per-friend "pings" option so that you get notified whenever certain friends check in. However, I've yet to see this work. I have pings turned on for my wife, but I never get an alert from my phone when she checks in anywhere.
I've checked Background Tasks, and Foursquare isn't even in there - as if there isn't even a service available for it to run in the background. Is there something I'm missing?