We study the redshift evolution of the X-ray and Sunyaev-Zel'dovich (SZ) scaling relations for galaxy groups and clusters in the FABLE suite of cosmological hydrodynamical simulations. Using an expanded sample of $27$ high-resolution zoom-in simulations, together with a uniformly-sampled cosmological volume to sample low-mass systems, we find very good agreement with the majority of observational constraints up to $z\sim 1$. We predict significant deviations of all examined scaling relations from the simple self-similar expectations. While the slopes are approximately independent of redshift, the normalisations evolve positively with respect to self-similarity, even for commonly-used mass proxies such as the $YX$ parameter. These deviations are due to a combination of factors, including more effective AGN feedback in lower mass haloes, larger binding energy of gas at a given halo mass at higher redshifts and larger non-thermal pressure support from kinetic motions at higher redshifts. Our results have important implications for cluster cosmology from upcoming SZ surveys such as SPT-3G, ACTpol and CMB-S4, as relatively small changes in the observable--mass scaling relations (within theoretical uncertainties) have a large impact on the predicted number of high-redshift clusters and hence on our ability to constrain cosmology using cluster abundances. In addition, we find that the intrinsic scatter of the relations, which agrees well with most observational constraints, increases at lower redshifts and for lower mass systems. This calls for a more complex parametrization than adopted in current observational studies to be able to accurately account for selection biases.