I am a body part in flames
The apple of one's eye, a phrase for describing something the person prizes. Eyes are also described as shining or bright when someone's enthusiastic, or alight or even burning with passion or certainty. The eyes burn, for the apple of one's eye.
The earth and I move the same
Maybe myths, like Johnny Appleseed or the garden of Eden, where apples are linked to people's relations to the earth. Maybe it is talking about growing seasons, and apple orchards - moving through the seasons (like the earth) to produce fruit.
I am a dessert without tea
Apple tea! Fruit by itself can be a dessert - an apple by itself, or with a fruit plate or a fruit and cheese platter or baked apples or a boiled apple pudding. Turkish apple tea is also quite popular and very good - but once tea has been added, it is a beverage and no longer a dessert.
When it comes to cake, I'm not easy
The phrase is 'easy as pie' - and apple pie is the quintessential pie, after all the phrase is 'as American as apple pie' for a reason. Alternatively, if the line is a little more literal, it might refer to apple-cake, of which there are many recipes, and which are tasty but not usually easy to balance.
Edit - apparently I took long enough answering to miss out on the hints... I'm probably not right, then. Ah, well.