No, not necessarily between tournaments or regular uses unless there is an extended length of time between uses. Depending on the type and age of the batteries, particularly alkaline cells, they can begin to leak when not in use and the acid that seeps out of them is highly corrosive. It's a good idea to remove batteries from any device if it will go unused for a significant length of time. If left unchecked it can ruin the device.
Game Clocks - REMOVE Those Batteries!

Or just use the lithium AA batteries: they weigh less, have 3 times the charge, have 20 year life before starting to drop charge AND they don't leak like alkaline ones do.
In fact use them in all the devices you have to have long stand by time.
Or use rechargeable (nickel metal hydride batteries) batteries for devices you use often. They also never leak are at 75% capacity after 5 years on the shelf and are... well rechargeable. Just make sure to get the ones made in japan. Any brand will do Panasonic, Eneloop, Duracell, Amazon, Energizer etc, If they are made in japan they all come from the same factory and are all great.
If you are not using your game clock during this period of social distancing, you might want to remove the batteries. Who knows how long it will be before we can play over the board and face-to-face again, so don't do like I have done with every flashlight I have ever owned. Remove those batteries and prevent corrosion to the electrical contacts and housing.