Time-resolved fluorescence spectroscopy is used to investigate the efficiency of Förster resonance energy transfer (FRET) between two different types of organic dye sensitizers bound to zirconia nanoparticles. The co-sensitization scheme involves using the IR125 dye as the FRET acceptor and either the D149 dye or D35 dye as the FRET donor, with the FRET parameters determined by monitoring the donor dye's excited state lifetime in the absence and presence of the acceptor dye. The FRET rate is found to be faster for the D149+IR125 pair than for the D35+IR125 pair. From the FRET quantum yields, the Förster distances for the D149+IR125 and D35+IR125 pairs are estimated to be 3.5 and 2.6 nm, respectively. Analysis of the data, in conjunction with time-dependent density functional theory calculations, leads to the conclusion that the variation in the Förster distances is dictated mainly by differences in the donor-acceptor orientation factors. For D149 the rhodanine acetic acid binding group constrains the dye to lie almost parallel to the surface, whereas the cyanoacrylic acid binding group of D35 causes the dye to sit almost perpendicular to the surface. Because IR125 lies parallel to the surface, due to the two sulfonate binding groups, there is better alignment of the transition dipole moments for the D149+IR125 pair than for the D35+IR125 pair. Deliberately choosing dyes with binding motifs that misalign the donor-acceptor transition dipole moments may be an effective strategy for suppressing FRET within dye-sensitized solar cells to enhance their efficiency.