Well, a bit more delicate than swapping out batteries.
You need to pay close attention to where Pin #1 is or at least recognize the notch in both the top edge of the chip, as well as the socket. Do NOT use direction of printing on the labels to determine which end goes where!!!!
You need to observe proper anti-static handling procedures so you don't either zap the EPROMs or any parts on the main MPU board.
And, you need to pay close attention to not bending any legs under as you only get so many "Bend backs" until they break off.
But most importantly, you need to make sure you don't go digging in with a flat head screw driver and have the tip hit traces under the socket. This is the most common mistake beginners make when trying to remove EPROMs from their sockets. If you are not using some kind of a chip puller, then with a flat head screw driver, make sure you gently pry between the chip and the socket (NOT the socket and the board) and as you push in, keep the screw driver parallel to the PCB so as not to be pointing down to slip and dig into the board below the socket. When you manage to get the screw driver between the chip and the socket, twist the screw driver to bring that end of the chip up a bit. Move to the other end of the EPROM and repeat. Go back and forth so you keep the end pins straight and not have to bend them back because you pried the chip up about 45 degrees before you lifted it out.