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Crystalline nanoporous molecular networks are assembled on the Ag(111) surface, where the pores confine electrons originating from the surface state of the metal. Depending on the pore sizes and their coupling, an antibonding level is shifted upwards by 0.1 to 0.3 eV as measured by scanning tunneling microscopy. On molecular sites, a down-shifted bonding state is observed, which is occupied under equilibrium conditions. Low-temperature force spectroscopy reveals energy dissipation peaks and jumps of frequency shifts at bias voltages, which are related to the confined states. The dissipation maps show delocalization on the supra-molecular assembly and a weak distance-dependence of the dissipation peaks. These observations indicate that two-dimensional arrays of coupled quantum dots are formed, which are quantitatively characterized by their quantum capacitances and resonant tunneling rates. Our work provides a method for studying the capacitive and dissipative response of quantum materials with nanomechanical oscillators.