We present Graph Neural ODEs (GNODEs) for modeling tumor microenvironment dynamics with mathematically guaranteed stability and conservation properties. Unlike bulk ODEs that miss spatial heterogeneity or discrete GNNs that inadequately capture continuous biological processes, GNODEs provide continuous-time evolution with explicit adjacency-aware dynamics while maintaining provable trajectory bounds. Our framework ensures: (1) existence and uniqueness of solutions under dynamic graph topology, (2) Lyapunov stability preventing unphysical states like negative cell counts, and (3) exact conservation of biological invariants through architectural constraints. Benchmarking on synthetic tumor data demonstrates that GNODE accurately captures the dynamics of the resistant cell fraction (0.282 predicted vs 0.242 true), whereas graph-free alternatives fail completely (0.000), underscoring the importance of stability-constrained local interactions for modeling emergent resistance.