This paper aimed to give some partial answers to the zero-product problem and commutativity problem concerning dual Toeplitz operators with nonharmonic symbols on the orthogonal complement of the harmonic Bergman space. Using the symbol map, we described the necessary condition for $S_{{\phi }_1}{S}_{{\phi }_2}\cdots S_{{\phi }_N}=0$ with radial symbols. Furthermore, we established the sufficient and necessary conditions for $S_{\phi }{S}_{\psi }=S_{\psi }{S}_{\phi }$ with $\phi (z)=a{z}^{p_1}{\overline{z}}^{q_1}+b{z}^{p_2}{\overline{z}}^{q_2}$ and $\psi (z)=z^s{\overline{z}}^t$.