When you really don't feel like eating but are guilted into eating. Eating, in order to "fit in".
"Oh, you better at least have a "polite bite" of your Birthday Cake!". Second example; When your Mom or someone offers you something to eat and you've recently eaten and just couldn't eat another bite. "Oh, honey just have a "polite bite", I made it just for you!". When someone insists you taste something and it doesn't look appealing to you. So in order to not offend them, you take a "polite bite".