  We investigate the dynamical properties of one-dimensional dissipative
Fermi-Hubbard models, which are described by the Lindblad master equations with
site-dependent jump operators. The corresponding non-Hermitian effective
Hamiltonians with pure loss terms possess parity-time ($\mathcal{PT}$) symmetry
if we compensate the system additionally an overall gain term. By solving the
two-site Lindblad equation with fixed dissipation exactly, we find that the
dynamics of rescaled density matrix shows an instability as the interaction
increases over a threshold, which can be equivalently described in the scheme
of non-Hermitian effective Hamiltonians. This instability is also observed in
multi-site systems and closely related to the $\mathcal{PT}$ symmetry breaking
accompanied by appearance of complex eigenvalues of the effective Hamiltonian.
Moreover, we unveil that the dynamical instability of the anti-ferromagnetic
Mott phase comes from the $\mathcal{PT}$ symmetry breaking in highly excited
bands, although the low-energy effective model of the non-Hermitian Hubbard
model in the strongly interacting regime is always Hermitian. We also provide a
quantitative estimation of the time for the observation of dynamical
$\mathcal{PT}$ symmetry breaking which could be probed in experiments.
