In this paper, we give the notion of F-biharmonic maps, which is a generalization of biharmonic maps. We derive the first variation formula which yields F-biharmonic maps. Then we investigate the harmonicity of $F$-biharmonic maps under the curvature conditions on the target manifold (N; h). We also introduce the stress $F$-bienergy tensor $S_{F;2}$. Then by using the stress F-bienergy tensor $S_{F;2}$, we obtain some non existence results of proper $F$-biharmonic maps under the assumption that $S_{F;2} = 0$. Moreover, we derive some monotonicity formulas for the special case of biharmonic map, i.e. $F$-biharmonic map with $F(t) = t$. Then, by using these monotonicity formulas, we obtain new results on the non existence of proper biharmonic isometric immersions from complete manifolds.