That responsibility is a burden to him.
Understandable but robotic unnatural translation… My alternative translation is:
身に余る in this context means “too big/important to properly handle”. It literally means “to exceed one’s body”. Please note that 身に余る can be used also in a positive context such as 身に余るお言葉を頂き、恐縮です。(I am very honored to have such a nice word from you.) So. 身に余る itself isn’t a negative phrase.
重責 means “pivotal role” or “critical responsibility”. 重責を担う (～をになう; to play a pivotal role) and 重責を果たす (～をはたす; to fulfill an important mission) are frequent terms in business.