Do I have to eat?

The food meter becomes relevant only when it becomes red and you begin to starve.
You won't restore HP or energy when resting, until you find some food to fill your food bar.
Healing is only a temporary solution, but it helps you gain time.
Other than that, no penalties.

EDIT: I don't know if this has been changed recently, but in the current version a starving character has his Attack Power halved(rounded up). This changes the strategies, since this way the hungry PC becomes almost useless, making a food reserve a necessity.

As the tooltip on the food bar indicates:

Yellow bar means your character is hungry. There are no penalties at this stage. This is the best time to eat food so you don't waste any (different food items restore the food bar by different amounts, but any over the maximum goes to waste).

Red bar means your character is starving and won't regenerate.

Food is consumed by actions (moving, casting spells, swinging weapons, levers, etc.) and during resting.