No. In fact, most volunteers speak only English. It is not a requirement and it’s completely possible to get by relying solely on the English language, friendly locals you meet, Google translate, and even some charades at times.
Living in another country is a unique opportunity to learn the local language, though. A lot of volunteers make an effort to learn phrases that they can use to communicate better with their new neighbors and friends. Even basic greetings can be a great way to show respect for the culture you’re in.