Thats very odd, mine doesn't turn off. I've locked it while it was downloading a big app (350 MB) and when I came back an hour later it was done. You sure it's actually turning off? Try locking it while downloading an app and see what happens.
Again, I don't think that the actual issue is the iPad, rather the router. However, if you don't want to mess around with the router, the easiest thing to do is restore the iPad and see if that's what's causing it.