Why eat when there was no death? In Genesis, God tells Adam and Eve not to eat of a certain tree, but they can eat from any other tree. The bible also apparently says that before sin there was no death.

So... Why the need to eat? If you are immortal, what purpose would eating have?
There are really only two opinions on this and they stem from differing definitions of the word immortal which often conflates or coincides with opinions on the nature of the afterlife.

Traditionally, immortal meant that you simply did not age, and therefore would not die from old age. This is often extended to illness as well. Your body still had requirements, such and food, water, and rest. Naturally, then, there was fruit in the garden for Adam to eat. Proponents of this view often point out that the Tree of Life was in the Garden too. They suggest that they Tree was necessary for immortality and was eaten regularly. This leave the question of harm to your body. Yes, that would be possible under this view.

In much of Christianity, immortal has come to mean that the person cannot die by any cause. The doctrines that dictate there is an immortal soul that is separate from the body use the word immortal in this sense. The pre-fall existence is often conflated with this, despite it being a significantly different thing. So they say the body had no requirements to sustain itself. Proponents of this view would likely point out that eating is a pleasurable exercise, hence, there was fruit in the Garden.

Ancient Greek religion originally tended more toward the first definition, but moved toward the second later. It had a significant impact on Christianity while the second definition was prominent. More precisely, it had an impact on the Jews of that time.