The purpose of the present article is to find solutions to a system of partial differential equations that characterize infinitesimal holomorphically projective transformations on the tangent bundle with the Sasaki metric and an adapted almost complex structure. Moreover, it is proved that if the tangent bundle of a Riemannian manifold admits a non-affine infinitesimal holomorphically projective transformation, then the Riemannian manifold is locally flat.