Not if you move the filter media, along with the fish from the 10 gallon. That media has enough bacteria to support the bio load of the fish in the 10 gallo, whether it be 10 gallons of water or 100. The tank is just a glass box that contains the water.
You can move the filter itself, or place the media from the 10 gallon filter into a larger filter. If you have only undergravel filtration, move all the gravel, as that it your bio media.