Agreed that removing the rims off your tank is not recommended, though this can vary from tank to tank brands but generally speaking these rims, either plastic or glass types, as Eagles pointed out, are structural to the integrity of your tank.
The rims help prevent the glass from coming apart at the edges/joints as also in large tank helps to stop the glass from bowing under the pressure of water. Water carries a good deal of weight and the subtrate and plants etc all add to the overall weight, so I'd keep the rim on the tanks for peace of mind.
Aside from the loss of your potential stocking, have you ever seen 10 gallons of water on the floor? Its more than you think and creates havoc on your carpet, laminate or wooden floors etc if not dried out properly. It's not pretty and can be a lot of work to sort out.