We measure superfluid transport of strongly interacting fermionic lithium atoms through a quantum point contact with local, spin-dependent particle loss. We observe that the characteristic non-Ohmic superfluid transport enabled by high-order multiple Andreev reflections transitions into an excess Ohmic current as the dissipation strength exceeds the superfluid gap. We develop a model with mean-field reservoirs connected via tunneling to a dissipative site. Our calculations in the Keldysh formalism reproduce the observed nonequilibrium particle current, yet do not fully explain the observed loss rate or spin current.