Why do fully grown organisms need protein intake?
Protein is polymerized amino acids.  

Humans do not require protein intake, but certain essential amino acids are required.  These essential amino acids are usually obtained through protein intake, but intake of the free amino acids themselves would suffice.  

Amino acids, in free or protein form, are lost from the body due to urination (see Diverse characteristics of the urinary excretion of amino acids in humans and the use of amino acid supplementation to reduce fatigue and sub-health in adults), perspiration, menstruation, ejaculation, tear drops (see Amino Acid profiles in human tear fluids ), skin exfoliation, hair and nail growth, nasal mucus (see Identification of human nasal mucous proteins using proteomics), and vaginal mucus (see Proteins of Cervical Mucus).  Additional loses can occur due to additional bleeding, lactation, child birth, spitting (and other activities causing loss of saliva), and coughing up mucus.  

Synthesis of new proteins from amino acids is constantly required by the body (see NUTRITION AND PROTEIN TURNOVER IN MAN). Though there are chemical pathways to reuse amino acids, it is necessary for the concentration of each amino acid to be non-zero, and therefore some fraction is necessarily lost through excretion.  

Beyond just direct excretion of amino acids and proteins, there is excretion of nitrogen-containing metabolites such as urea, uric acid, creatinine, ammonium salts and bilirubin.