What resolution/dpi should I scan a photograph at if I want to use it on a product?
It depends on how large you want the image to be.
A good rule of thumb is to scan it at twice the resolution you think you'll need. For Cafepress products the largest printable area we've been given is 10" x 8" (or 8" x 10") this is at a suggested 100 dpi.
If you intend to add the photograph onto mugs ,though you will need it at a minimum of 200 dpi.
For simplicity sake let's say you want the photograph to come out at 100% of it's original size. You can then scan it at 100 dpi for use on the shirt. This, however will mean it will only be 50% of it's original size on the mug.
To save time do this instead: scan the photograph at 400 dpi. This will result in a much bigger size but it will cover any printing area needed. You can always reduce the finished scan in size to fit an item. But it is not advisable to enlarge a scan as the data will be interpolated then and not be crisp. (Fricka)