Boron nitride nanotubes (BNNTs), a type of nanomaterial that was first made in 1995, have gained attention in the last two and a half decades, having been characterized as promising inorganic nanoparticles for the application and development of devices in nanomedicine (such as drug delivery systems), and chemical sensors, among other applications. With the use of density functional theory (DFT) calculations, we study the effect of functionalizing the surface of a BNNT with hydroxyl (-OH) and carboxyl (-COOH) organic groups at the N and B sites. The results indicate that the -OH radical remains adsorbed in the B site whereas it is detached from the surface when added to the N site. Very differently, the -COOH remains attached for both adsorption sites. Both functionalizations induced a magnetic moment, produced spin-polarized electronic bands, created flat impurity-like bands, decreased the band gaps, and modified the electron density of the nanotubes. All these changes can be used to control and modify the interaction of BNNTs with other molecules of interest like drugs, heavy metals, and greenhouse gasses, among other substances.