Do GBA cartridge batteries run dry after some time?

Solution 1:

Only certain Game Boy Advanced cartridges use a battery backup. A GBA cartridge can contain either EEPROM, SRAM, or Flash RAM. Only the SRAM variant would need a battery backup. So for most games it is not a problem.

The only games I know of that use battery power to hold save data are as follows:

• Baseball Advance
• Boktai 1, 2 and 3 (JP)
• Bomberman Tournament
• Breath of Fire
• Breath of Fire II
• Castlevania: Circle of the Moon
• Castlevania: Harmony of Dissonance
• Dragonball Z Collectible Card Game
• Hamtaro: Ham-Ham Heartbreak
• Kirby: Nightmare in Dream Land
• Krazy Racers
• Lego Bionicle
• Lego Island 2: The Brickster's Revenge
• MegaMan Battle Network
• MegaMan Zero
• Metroid Fusion (Only some use a battery)
• Metroid Zero Mission
• Monster Rancher Advance
• Monster Rancher Advance 2
• Pokemon Emerald (Not for saving, but to power the Real Time Clock)
• Pokemon Ruby (Not for saving, but to power the Real Time Clock)
• Pokemon Sapphire (Not for saving, but to power the Real Time Clock)
• Super Dodgeball Advance
• Warioland 4

Source GBAtemp.

Solution 2:

Yes, this will definitely happen. No battery lasts forever. Eventually, you will need to replace them in order to be able to store game saves.