Is it permissible to broadcast a religious program that includes musical interludes (intros/outros) of a religious nature, and would I incur sin if I alerted viewers to mute the sound when music is present, and they failed to do so?

It is not permissible to publish programs that contain musical clips, especially given the availability of alternatives free from Sharia prohibitions. Notifying users about the presence of music and requesting them to mute the sound is insufficient when publishing them to the general public, as it is highly probable that many will not adhere to this notification. This is only permissible when sending it to a specific, trustworthy individual.