Does the DVR allow you to burn DVD-R's? (unlikely, but some do)
You could also pull the hard drive out and try reading it with a USB adapter (but you might have to break a seal on the DVR, and there's no guarantee the file will be in a nonproprietary format.
Or you could find someone with a DVR that *does* have a DVD burner...
Check with your local TV stations, they may can get a video clip direct from the network.