Plate-type acoustic metamaterials are typically constructed from a tensionless membrane. These acoustic metamaterials are characterized entirely based on the resonant behavior of the membrane. To enhance the acoustical performance, an additional feature such as a tonraum resonator can be incorporated into the design. Consequently, more design parameters can be tuned on top of the material properties and the geometries of the membrane. However, the resonator's effect could be damped (i.e., missing sound transmission loss peak and dip) because of (1) the poor cylindricity of the orifice produced by fused deposition modeling, and (2) the viscous losses caused by the acoustic boundary layer developed within the orifice. This study clarifies the influence of the two possible causes on the damping of the resonator's effect by investigating the specimen from a previous work in greater detail numerically and experimentally. To achieve this objective, the shape and the depth of the orifice were varied. The results show that the damping of the resonator's effect was not caused by the poor cylindricity of the printed orifice, but was attributed to the viscous losses induced by the acoustic boundary layer developed within the orifice. As a side note, the observations from a comparative study show that the printing time of the specimen can be reduced up to 41% without compromising on acoustical performance by having a lower infill percentage. Consequently, this study offers complementary insights into the design of plate-type, and even membrane-type, acoustic metamaterials with an internal tonrawn resonator so that they can be tuned for low-frequency noise control applications if required and not be implicated by the viscous boundary layer developed within the orifice.