I want to remove tiles and grout from a table top so I can redo the tile. What is the best tool for doing this without damaging the surrounding wood table top?

An Angle grinder with a diamond blade. A Multitool, could do it , but would be slower.
If not proficient with power tools, they can be more Dangerous, an Example:
http://www.homedepot.com/Tools-Hardware/Makita/h_d1/N-5yc1vZc1xyZyg/R-202105761/h_d2/ProductDisplay?catalogId=10053&langId=-1&storeId=10051

An Example of the Blade :
http://www.amazon.com/Makita-A-94546-4-Inch-Diamond-Masonry/dp/B000MQQ81W

Got a sawzall?:
http://www.zorotools.com/g/00056130/k-G1654107?utm_source=google_shopping&utm_medium=cpc&utm_campaign=Google_Shopping_Feed&kw={keyword}&gclid=COzDsObYqbECFc2a7QodUQIAQg

You can buy a grout hand saw @HD, Lowe's or a tile place. Takes a little work. If you have a Multi tool you can buy a grout blade for it.

To remove tile u will have to break one tile out. U can then chisel the rest of tile out.

Chuck