Coriolis flowmeters determine the mass flow rate by measuring the phase shift or time delay between the signals from the pick-off coils. Therefore, any change in the phase shift unrelated to fluid flow contributes to the mass flow error. Error sources can include external influences and non-idealities in the flow tubes. This article describes an experimental study of the influence of one of these factors - the boundary condition stiffness of the Coriolis flowmeter - on the mass flow error. In the experiments, we obtained a dependence of the mass flow error on the boundary condition stiffness. The maximum error exceeded the allowable range by more than ten times. This corresponds to the case when the modal frequency of the 'Coriolis flowmeter-pipeline' system coincides with the drive frequency. It should be noted that the studied error is caused by the zero shift and can change over time, for example, when the temperature changes. Further, we developed a method of diagnosing random zero shift of the Coriolis flowmeter caused by the condition of its mechanical support. The method is based on measurement and analysis of modal damping. To validate the method, we carried out experimental studies to evaluate modal damping depending on the boundary condition stiffness of the flowmeter, the water flow rate and in the conditions of the water-air mixture flow. The results of the laboratory tests confirmed the practical applicability of this method.