Vector fields are widely used to represent and model flows for many science and engineering applications. This paper introduces a novel intrinsic neural network architecture for learning tangent vector fields defined on manifold surfaces embedded in 3D. Previous approaches to learning vector fields on surfaces treat vectors as multi-dimensional scalar fields, using traditional scalar-valued architectures to process channels individually, thus failing to preserve fundamental intrinsic properties of the vector field defined on the surface. The core idea of this work is to introduce a trainable vector heat diffusion module to spatially propagate vector-valued feature data across the surface, which we incorporate into our proposed architecture consisting of vector-valued neurons. Our architecture is invariant to rigid motion of the input and choice of local tangent bases, and is robust to discretizations of the surface. We evaluate our Vector Heat Network on one particular discrete manifold surface, triangle meshes, and we validate its invariant properties experimentally. We also demonstrate the effectiveness of our method on the useful industrial application of quadrilateral mesh generation.