Trello – subscribe to a trello board’s Activity list with something like Google Reader

Currently, you can only view open boards, so you'd need to re-open the archived board first.

Once you've re-opened the board, you should be able to

1. Click "Options"
2. Click "Archived Items"

I do not believe Trello has this functionality built in. If you're willing to use another service to get notifications, I recommend checking out this Trello to Google Calendar Zap on Zapier.

The way it works, whenever a card is updated with a due date, the Zap will automatically put the due date onto a Google Calendar. Google Calendar provides many mechanisms for upcoming event notifications including customized notifications, variable-length notifications periods, etc. You should be able to receive these on your mobile device, too!

I should note, to get this Zap to work, you'll need set up a filter (when you set up the Zap) to only trigger on cards and only those with due dates.

Alternatively, you might look at this Zap which is easier to set up, but requires you to immediately choose a due date (within a few minutes of making the card) else the due date won't exist when the Zap triggers.