I would guess that it is
The Earth.
Cut me for nothing to be missed,
Humans dig into the Earth to get to natural resources.
Open me to be surprised.
The Earth's core is theorized to be molten, which is surprising the first time you hear about it.
Be my fruit and you'll be loved,
The fruit of the Earth generally refers to the good things of the Earth, or the growing things, which people love and need.
Though if I'm from an animal, you'll be doomed.
I'm unsure. Some animals are territorial or have hunting grounds, so their parts of the Earth might be dangerous...? On the other hand, some people might consider you morally doomed if you took over land where an endangered species makes its home.
Edit
I hope I am permitted to give a second answer. I have another guess:
A seed.
Cut me for nothing to be missed,
Cutting seeds in a particular way (scarification) can speed germination, making it easier to sprout difficult-to-grow seeds.
Open me to be surprised.
A nut is a seed that you must open to get to the meat. Otherwise, plant a seed and it opens into a sprout.
Be my fruit and you'll be loved,
Many people "love" fruit, and fruit has seeds.
Though if I'm from an animal, you'll be doomed.
"Seed" is sometimes used to mean "offspring", such as in the Bible. Taking an animal's offspring away can be very risky, because the mothers will often be protective. This is especially true for mammals.
