Cut me for nothing to be missed,
open me to be surprised.
Be my fruit and you'll be loved,
Though if I'm from an animal you'll be doomed.
Who am I?
An eye?
Cut me for nothing to be missed
A slightly dated term - to cut one's eyes at something is to briefly look at it - thanks, @Cubicon!
open me to be surprised
How many surprises start with being led somewhere with your eyes closed?
Be my fruit and you'll be loved
She was the apple of her father's eye
Though if I'm from an animal you'll be doomed
Bullseye, if someone's aiming for you
My answer is
Egg
Cut me for nothing to be missed,
You break (cut) the egg if you want to eat the yolk.
open me to be surprised.
Chocolate eggs with a surprise inside are typical of Easter
Be my fruit and you'll be loved,
There's a fruit called Eggfruit
Though if I'm from an animal you'll be doomed.
Eggs come from animals. Doomed might be a reference to a word like Egged?
I think that you are
Amber, as used in Jurassic Park
Cut me for nothing to be missed,
They drilled, or cut, open the stone to extract what might have otherwise been missed.
open me to be surprised.
When they opened it up, they were surprised by how well preserved the fossil was.
Be my fruit and you'll be loved,
The Amber's "fruit", the mosquito, was treasured by the scientists for its value in research.
Though if I'm from an animal you'll be doomed.
Turns out the mosquito had eaten some dino blood. Spoiler alert! Resurrecting Dinosaurs may result in injury or death.
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.
Beehive.
Cut me for nothing to be missed,
To get all honey.
open me to be surprised.
See creatures from penta-hexa-gons
Be my fruit and you'll be loved,
Honey can be considered as fruit of beehive and it is lovable.
Though if I'm from an animal you'll be doomed.
Bees may doom someone.
I think this is some sort of
parasite.
"Cut me for nothing to be missed,"
Cut (remove) a parasite and you won't miss anything.
"open me to be surprised"
I have no idea how this relates.
"Be my fruit and you'll be loved,"
Parasites would consider their host to be a "fruit".
"Though if I'm from an animal you'll be doomed"
Many parasites that first attach to an animal host and later a human can transfer diseases picked up from said animals.
I think it's a
Dragon fruit (Pitaya)These fruits are commonly known in English as "dragon fruit"
Cut me for nothing to be missed,
You have to cut this fruit for you to eat the creamy and delicious inside
open me to be surprised.
If you open this fruit, you'll be surprised of its bright white flesh that looks similar to a kiwi inside, with tiny black seeds distributed throughout the fruit.
Be my fruit and you'll be loved,
This fruit is exotic and many people want to have this fruit because of the extreme health benefits that it gives
Though if I'm from an animal you'll be doomed.
If it becomes a real dragon, you'll be really doomed!