Autonomous systems operating in real-world conditions often have to contend with environmental disturbance behavior that is unknown a priori. We present a method for efficiently computing reachable sets for continuous-time systems with partially unknown dynamics. Our main assumption is that, given any hyperrectangle of states, lower and upper bounds for the unknown components are available. With this assumption, the theory of mixed monotone systems allows us to formulate an efficient method for computing hyperrectangular overapproximations of the reachable sets of the system. We apply this formulation to a system navigating towards a goal region while avoiding unsafe regions. We derive a model predictive control scheme that avoids the unsafe region and ensures the system is always within reach of an a priori guaranteed safe region, thus ensuring feasibility until the goal is reachable. We explore this formulation further by considering multiple probability levels to increase performance. We also consider the problem of tracking a reference trajectory for these systems. We modify the embedding system such that a single controlled trajectory corresponds to a controlled forward invariant interval tube around the reference, and utilize it in a runtime assurance mechanism that guarantees tracking of the reference trajectory within a desired threshold.