Are elimination reactions only organic reactions?
No, eliminations are also biochemical reactions.
The difference is that they are catalyzed by enzymes, not by strong acids and bases.
They also tend to be E1cB eliminations, in which a base first abstracts an acidic proton to generate a stabilized anion.
For example, argininosuccinate lyase is an enzyme in the urea cycle. This is a series of reactions in which the amine group of amino acids is transferred to urea for elimination in the urine.
It catalyzes an anti-elimination from argininosuccinate. Arginine is the leaving group, and fumarate is the other product.
In fatty acid synthesis, the growing fatty acid chain is attached to an acyl carrier protein (ACP).
β-HydroxyacylACP dehydrase is an enzyme involved in fatty acid synthesis. It catalyzes the elimination of water from β-hydroxyacylACP esters to form the corresponding α,β-unsaturated acylACP esters.