The maximum independent set problem is a classical NP-hard problem in theoretical computer science. In this work, we study a special case where the family of graphs considered is restricted to intersection graphs of sets of axis-aligned hyperrectangles and the input is provided in an online fashion. We prove results for several adversary models, classes of hyperrectangles, and restrictions on the order of the input. Under the adaptive offline and adaptive online adversary models, we find the optimal online algorithm for unit hypercubes, $σ$-bounded hypercubes, unit-volume hyperrectangles, and arbitrary hypercubes, in both non-dominated and arbitrary order. Under the oblivious adversary model, we prove bounds on the competitive ratio of an optimal online algorithm for the same classes of hyperrectangles and input orders, and we find algorithms that are optimal up to constant factors. For input in dominating order, we find the optimal online algorithm for arbitrary hyperrectangles under all adversary models. We conclude by discussing several promising directions for future work.