(1) Background: Healthcare workers are highly affected by work-related musculoskeletal disorders, particularly in the lower back, neck and shoulders, as their occupational tasks expose them to biomechanical constraints. One solution to prevent these musculoskeletal disorders may be the use of a passive exoskeleton as it aims to reduce muscle solicitation. However, few studies have been carried out directly in this field to assess the impact of the use of a passive upper limb exoskeleton on this population. (2) Methods: Seven healthcare workers, equipped with electromyographic sensors, performed a tool cleaning task with and without a passive upper limb exoskeleton (Hapo MS, Ergosanté Technologie, France). Six muscles of the upper limbs were analysed, i.e., anterior deltoid, biceps brachii, pectoralis major, latissimus dorsi, triceps brachii and longissimus thoracis. A subjective analysis of the usability of the equipment, the perception of effort and discomfort, was also carried out using the System Usability Scale and the Borg scale. (3) Results: The longissimus thoracis was the most used muscle during this task. We observed a significant decrease in the muscular solicitation of the anterior deltoid and latissimus dorsi when wearing the exoskeleton. Other muscles were not significantly impacted by the device. (4) Conclusions: the passive exoskeleton used in this study allowed the reduction in muscular load on the anterior deltoid and latissimus dorsi without negative effects on other muscles. Other field studies with exoskeletons are now necessary, particularly in hospitals, to increase our knowledge and improve the acceptability of this system for the prevention of musculoskeletal disorders.