Connecting to an EEPROM
Update

Here is a good list of micro grabbers/test clips (or whatever you want to call them): http://www.jameco.com/jameco/products/prodds/684481-ds01.pdf


As mentioned, it would be helpful if we knew the board and the chip. But without that information here is what I recommend.

First off, you need to buy some micro grabbers. These will allow you to connect to individual pins on medium sized surface mount components (usually around .5mm pitch). They look like this:



Now if you power to board and listen in on the bus, you can gather needed information like i2c address, etc. and see how active it is. From here you may be able to dump the memory without any ill effects. Otherwise you can use my method for removing surface mount parts to only remove the data communication pins. Place something non-conductive between the pin and the pcb board so it won't make contact. Then read out the memory like you planned. This will make it much easier for you to resolder the two or so pins you removed back on, and is easier on the chip than completely removing it and trying to power it, etc.

If you don't have or want to get micro-grabbers, you can very carefully place a ~22 gauge solid wire under the pin but on top of the non-conductive material, so that the wire is making contact with the pin. This will be hard to keep from disconnecting but with a bit of work you will get the hang of it.

If at all possible, I would stay away from soldering wires directly to the pins!