Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices, and they are interesting in their own right, with ordinary percolation exhibiting not one but two phase transitions. We study four constraint percolation models-k-core percolation (for k=1,2,3) and force-balance percolation-on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggest that all of the k-core models, even for k=3, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the k-core percolation models.