Scraping
If the item you are trying to remove the tape from is flat and hard, try using a sharp razor blade held flat to the surface to gently scrape up a corner of the tape. If the tape is brittle and tears off when you try to pull it up, again scrape up a corner and try to pick it off a little at a time until it is removed. If the blade gets gummed up, carefully wipe the edge clean to keep it sharp.
Paper
Scotch tape should never be used to repair torn pages in books, as any book collector who has tried to restore a valuable first edition will know. Removing the tape from a book page after so much time and pressure on it can be nearly impossible without damaging the paper further. It may be possible to scrape off by holding a razor blade vertically to the paper and "shaving" it off, with a lot of patience. If the tape is on a glossy page like a book cover (not a photograph), scraping and a light application of rubbing alcohol with a cotton swab on the edges may do the trick.
Removing Adhesive
After the tape has been removed, you can attempt to take off any residual adhesive from the item. In many cases, it can be softened up with rubbing alcohol, or a similar commercial cleaning product, then scraped or rubbed off. Adhesive residue on photographs and similar fragile items should only be rubbed off, or damage could result from the use of cleaner. To rub, you can simply use your fingertip and rub in one direction until it balls up or reaches the edge. You can also use something like a toothbrush dipped lightly in a cleaning solution.
Cleaning Solution
You can make an effective cleaning solution simply by using warm soapy water and a small capful of ammonia. Straight rubbing alcohol works well too, but may need to be diluted if the item is fragile or sensitive. Many commercial cleaning products are designed for specialized or professional quality restoration, as well. Use whatever fits the project to apply, such as washcloth, cotton swap, toothbrush or sponge. The key is to be patient and as gentle as possible.