"The internal battery has run dry. Time-based events will no longer occur." That message came on my Emerald, but it didn't seem to affect my game at all. But what can happen: Eevee happiness at day = no Espeon. Eevee happiness at night = no Umbreon. (Both of these are based on the clock in your room. Any other day/night evolves won't work either). Berries might not grow like normal. They will be buried and like...well, nothing. Also, Eggs might not be created. Yeah, not good. Just because one of these happens doesn't mean all of them will. Lastly, you might not be able to return to saved data. Although that's not very likely, it might be good to take some precautionary measures. You might want to migrate your favorite Pokemon to Diamond/Pearl/Platinum, or trade them to any other Gameboy game. Those are the effects of a dry battery. Sorry it happened to you. Last thing: Since I already have Ruby and Sapphire, I gave my sister Emerald and let her delete my account. But for some reason, whenever she turns it on, the dry-battery message is gone. Now, I'm not suggesting you should delete your account, but...I just thought it was interesting and I should probably put it in.
Wow, pretty long answer.