We characterize Banach lattices on which each positive weak* Dunford--Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $F$ is a Banach lattice with order continuous norm, then each positive weak* Dunford--Pettis operator $T : Elongrightarrow F$ is weakly compact if, and only if, the norm of $E^{prime}$ is order continuous or $F$ is reflexive. On the other hand, when the Banach lattice $F$ is Dedekind $sigma$-complete, we show that every positive weak* Dunford--Pettis operator $T: Elongrightarrow F$ is M-weakly compact if, and only if, the norms of $E^{prime}$ and $F$ are order continuous or $E$ is finite-dimensional.