Not necessarily, there's already a web component to Xbox Live at xbox.com that you can login to from any browser and you can even do things on it (today) like reset all your Xbox 360 digital content licenses (I think this is limited to once per year) for exactly this sort of situation where your console died and you had to buy a new one.
So, log in to xbox.com, see list of your active licenses, click 'deactive' link on the games you want to forfeit your current digital license for. No console required.
If your console allows you to play disc purchased games without the disc in the console, then being able to "deactivate" the disc on the web before reselling it would allow you to continue to play your game on your un-networked console.
You can't have disc-less disc games that allow trading and don't require some form of "always on". You have to give up one of those; this time around it seems they are giving up "disc-less disc".
So, log in to xbox.com, see list of your active licenses, click 'deactive' link on the games you want to forfeit your current digital license for. No console required.