Continuous queries are often employed to monitor the locations of mobile objects (MOs), which are determined by sensing devices like GPS receivers. In this paper, we tackle two challenges in processing continuous range queries (CRQs): coping with data uncertainty inherently associated with location data, and reducing the energy consumption of battery-powered MOs. We propose the concept of spatiotemporal tolerance for CRQ to relax a query's accuracy requirements in terms of a maximal acceptable error. Unlike previous works, our definition considers tolerance in both the spatial and temporal dimensions, which offers applications more flexibility in specifying their individual accuracy requirements. As we will show, these tolerance bounds can provide well-defined query semantics in spite of different sources of data uncertainty. In addition, we present efficient algorithms that carefully control when an MO should sense or report a location, while satisfying these tolerances. Thereby, we particularly reduce the number of position sensing operations substantially, which constitute a considerable source of energy consumption. Extensive simulations confirm that the proposed algorithms result in large energy savings compared to nontolerant query processing.