We present the barycentric Lagrange dual tree traversal (BLDTT) fast summation method for particle interactions. The scheme replaces well-separated particle-particle interactions by adaptively chosen particle-cluster, cluster-particle, and cluster-cluster approximations given by barycentric Lagrange interpolation at proxy particles on a Chebyshev grid in each cluster. The BLDTT is kernel-independent and the approximations can be efficiently mapped onto GPUs, where target particles provide an outer level of parallelism and source particles provide an inner level of parallelism. We present an OpenACC GPU implementation of the BLDTT with MPI remote memory access for distributed memory parallelization. The performance of the GPU-accelerated BLDTT is demonstrated for calculations with different problem sizes, particle distributions, geometric domains, and interaction kernels, as well as for unequal target and source particles. Comparison with our earlier particle-cluster barycentric Lagrange treecode (BLTC) demonstrates the superior performance of the BLDTT. In particular, on a single GPU for problem sizes ranging from $N$=1E5 to 1E8, the BLTC has $O(N\log N)$ scaling, while the BLDTT has $O(N)$ scaling. In addition, MPI strong scaling results are presented for the BLTC and BLDTT using $N$=64E6 particles on up to 32 GPUs.