We investigate the stellar mass and baryonic mass Tully-Fisher relations (TFRs) of massive star-forming disk galaxies at redshift z~2.3 and z~0.9 as part of the KMOS^3D integral field spectroscopy survey. Our spatially resolved data allow reliable modelling of individual galaxies, including the effect of pressure support on the inferred gravitational potential. At fixed circular velocity, we find higher baryonic masses and similar stellar masses at z~2.3 as compared to z~0.9. Together with the decreasing gas-to-stellar mass ratios with decreasing redshift, this implies that the contribution of dark matter to the dynamical mass at the galaxy scale increases towards lower redshift. A comparison to local relations reveals a negative evolution of the stellar and baryonic TFR zero-points from z=0 to z~0.9, no evolution of the stellar TFR zero-point from z~0.9 to z~2.3, and a positive evolution of the baryonic TFR zero-point from z~0.9 to z~2.3. We discuss a toy model of disk galaxy evolution to explain the observed, non-monotonic TFR evolution, taking into account the empirically motivated redshift dependencies of galactic gas fractions, and of the relative amount of baryons to dark matter on galaxy and halo scales.