We present the stellar mass (M*) and Wide-Field Infrared Survey Explorer absolute Band 1 magnitude (M-W1) Tully-Fisher relations (TFRs) of subsets of galaxies from the CO Legacy Database for the GALEX Arecibo SDSS Survey (COLD GASS). We examine the benefits and drawbacks of several commonly used fitting functions in the context of measuring CO(1-0) linewidths (and thus rotation velocities), favouring the Gaussian Double Peak function. We find the M-W1 and M-* TFR, for a carefully selected sub-sample, to be M-W1 = (-7.1 +/- 0.6)[log (W-50/sin i/km s(-1)) - 2.58] - 23.83 +/- 0.09 and log (M-*/M-circle dot) = (3.3 +/- 0.3)[log (W-50/sin i/km s(-1)) - 2.58] + 10.51 +/- 0.04, respectively, where W-50 is the width of a galaxy's CO(1-0) integrated profile at 50 per cent of its maximum and the inclination i is derived from the galaxy axial ratio measured on the Sloan Digitized Sky Survey r-band image. We find no evidence for any significant offset between the TFRs of COLD GASS galaxies and those of comparison samples of similar redshifts and morphologies. The slope of the COLD GASS M-* TFR agrees with the relation of Pizagno et al. However, we measure a comparatively shallower slope for the COLD GASS M-W1 TFR as compared to the relation of Tully & Pierce. We attribute this to the fact that the COLD GASS sample comprises galaxies of various (late-type) morphologies. Nevertheless, our work provides a robust reference point with which to compare future CO TFR studies.