Human red blood cells (RBCs) need to deform in order to pass through capillaries in human vasculature with diameter smaller than that of the RBC. An altered RBC cell membrane stiffness (CMS), thereby, is likely to have consequences on their flow rate. RBC CMS is known to be affected by several commonly encountered disease conditions. This study was carried out to investigate whether an increase in RBC CMS, to the extent seen in such commonly encountered medical conditions, affects the RBC flow rate through channels with diameters comparable to that of the RBC. To do this, we use RBCs extracted from a healthy individual with no known medical conditions and treated with various concentrations of Bovine Serum Albumin (BSA). We study their flow through polycarbonate membranes with pores of diameter 5μm and 8μm which are smaller than and comparable to the RBC diameter respectively. The studies are carried out at constant hematocrit and volumetric flow rate. We find that when the diameter of the capillary is smaller than that of the RBC, the flow rate of the RBCs is lowered as the concentration of BSA is increased while the reverse is true when the diameter is comparable to that of the RBC. We confirm that this is a consequence of altered CMS of the RBCs from their reorientation dynamics in an Optical Tweezer. We find that a treatment with 0.50mg/ml BSA mimics the situation for RBCs extracted from a healthy individual while concentrations higher than 0.50mg/ml elevate the RBC CMS across a range expected for individuals with a condition of hyperglycemia. Using a simple theoretical model of the RBC deformation process at the entry of a narrow channel, we extract the RBC membrane bending modulus from their flow rate.