When we move, our ability to detect tactile events on the moving limb is reduced (e.g., movement-related tactile suppression). This process prevents unimportant sensory information from bombarding our central nervous system. This study investigated whether movement-related suppression can be modulated according to task relevance, while introducing a novel motor-driven complex upper limb movement. In three experiments, participants performed volitional self-driven and passive motor-driven reaching and grasping movements. Over the course of the movement, weak electrical stimulation was presented at task-relevant (i.e., index finger) and irrelevant sites (i.e., forearm) on the moving limb. In Experiment 1, participants displayed reduced detectability during movement (90% resting detection). This was true for all locations on the moving limb irrespective of task-relevance and during both self and motor-driven movements. In Experiments 2 and 3 a range of stimulus amplitudes were presented to one task-relevant location during both self and motor-driven movements (Experiment 2A), to a task-relevant and irrelevant site (Experiment 2B) and during a targeted and pantomime/no target reach (Experiment 3). This allowed us to estimate perceptual thresholds and assess the magnitude of movement-related suppression. During both self and motor-driven movements participants exhibited movement-related suppression. Suppression was greater at the irrelevant site (forearm) than at the relevant site (index finger) of the limb. Further, the magnitude of suppression varied with task relevance such that pantomime movements elicited more suppression than targeted movements. Collectively, these experiments suggest that although tactile suppression may be a general consequence of movement, suppression can be modulated in a relevance-dependent manner.