Hacham Ovadia Yosef, in his work Hazon Ovadia – Ta’aniyot, addresses the question of whether a professional music teacher may continue giving music lessons during the three weeks from Shiba Asar Be’Tamuz through Tisha B’Ab. As we know, it is forbidden to listen to or play music during this period, and the question thus arises as to whether an instructor may teach students how to play instruments, such as a guitar or piano, during this period. Hacham Ovadia rules that if a person earns a living by giving music lessons, he may continue his work during the Three Weeks, until the week of Tisha B’Ab, in order to avoid a loss of income. Once the week of Tisha B’Ab begins, he should refrain from teaching music.
On the basis of this ruling, we may conclude that during the Omer period, too, when music is generally forbidden, a professional music teacher may give music lessons. Just like during the Three Weeks, a teacher who will lose income by refraining from giving music lessons may continue giving lessons during the period of Sefirat Ha’omer. This is, indeed, Hacham Ovadia’s ruling in Yalkut Yosef (Hilchot Sefirat Ha’omer, p. 73; listen to audio recording for precise citation).
Summary: A teacher who will lose income by refraining from giving music lessons may continue giving lessons during the period of Sefirat Ha’omer, even though playing and listening to music is generally forbidden during this period.