Primarch essence? And it doesn't transform Marines into warp beings?
Gene Seed is derived from the Primarchs.
Is it an organ? A juice bag that occasionally needs refilling?
Gene Seed refers to the nineteen organs (tweny-one for Primaris) implanted into human aspirants that transforms them into Space Marines. These include;
Secondary Heart: another heart
Ossmodula: enhances bones
Biscopea: enhances muscle
Haemastamen: enhances blood
Larraman's Organ: allows for almost instantaneous healing
Catalepsean Node: allows a Space Marine to operate for as long as two weeks without sleep
Preomnor: detoxifies food before digestion
Omophagea: allows a Space Marine to gain knowledge by eating brain tissue
Multi-lung: another lung
Occulobe: enhances vision
Lyman's Ear: enhances hearing
Sus-an Membrane: allows a Space Marine to enter suspended animation in response to grievous wounds
Oolitic Kidney: detoxifies blood
Neuroglottis: enhances taste
Mucranoid: enhances skin
Betcher's Gland: allows a Space Marine to spit acid
Progenoid Glands: two glands that when matured can be extracted to create two new sets of Gene Seed organs.
Black Carapace: allows a Space Marine to interface with his Power Armour as if it were a second layer of skin.
Primaris Marines possess three additional Gene Seed organs;
Sinew Coils: further enhances muscle strength to the point of acting like subdermal armour
Magnificat: further increases body size as well as enhancing other Gene Seed organs
Belisarian Furnace: releases performance enhancing chemicals that also aid in regeneration, activated during extreme stress or trauma
What happens to a Marine if it is removed while he's alive?
Nothing. However the procedure is very invasive and risky due to how close the Progenoid Glands are to other vital organs so they usually wait until the Astartes is already (nearly) dead before extracting his Gene Seed.
What happens to all of the geneseed left in dead bodies on major battlefields?
Those are eventually recovered assuming that the bodies are still intact.