Hierarchical $\mathcal{H}^2$-matrices are asymptotically optimal representations for the discretizations of non-local operators such as those arising in integral equations or from kernel functions. Their $O(N)$ complexity in both memory and operator application makes them particularly suited for large-scale problems. As a result, there is a need for software that provides support for distributed operations on these matrices to allow large-scale problems to be represented. In this paper, we present high-performance, distributed-memory GPU-accelerated algorithms and implementations for matrix-vector multiplication and matrix recompression of hierarchical matrices in the $\mathcal{H}^2$ format. The algorithms are a new module of H2Opus, a performance-oriented package that supports a broad variety of $\mathcal{H}^2$-matrix operations on CPUs and GPUs. Performance in the distributed GPU setting is achieved by marshaling the tree data of the hierarchical matrix representation to allow batched kernels to be executed on the individual GPUs. MPI is used for inter-process communication. We optimize the communication data volume and hide much of the communication cost with local compute phases of the algorithms. Results show near-ideal scalability up to 1024 NVIDIA V100 GPUs on Summit, with performance exceeding 2.3 Tflop/s/GPU for the matrix-vector multiplication, and 670 Gflops/s/GPU for matrix compression, which involves batched QR and SVD operations. We illustrate the flexibility and efficiency of the library by solving a 2D variable diffusivity integral fractional diffusion problem with an algebraic multigrid-preconditioned Krylov solver and demonstrate scalability up to 16M degrees of freedom problems on 64 GPUs.