Eating (and sleeping) generally relate to your stamina. If you continually let your stamina drop but not eating AND not sleeping, or sleeping late (after 11 pm), you will start to notice adverse effects. Stormy days burn your stamina faster, so making sure to eat on those days can help you accomplish more in a day.
If your stamina drops during the day, and you don't eat, you will eventually grow tired and if you let it drop completely, you will pass out. When you get to that point, you will notice that your health drops, and you will have to make sure to take care of yourself (by eating and sleeping) in order to build it back up again.
If you generally avoid using most of your stamina in a day, and manage to sleep early on most days, you won't find that there are any adverse effects to not eating.