The other charities to be assessed will be the Young Concert Artists Trust, which supports young classical musicians, the Castle Players, a community theatre organisation from County Durham, and the Gwent Ballet Theatre, also known as Independent Ballet Wales.
A commission spokeswoman said the regulator wanted to assess arts organisations of varying sizes and scope. She said the assessments would probably take between six and nine months to complete.
The Royal Opera House volunteered to be assessed after being involved in drawing up the commission's guidance on public benefit and fee charging.
The first round of public benefit assessments, which ended in July, examined 12 religious charities, fee-charging schools and care homes. It found that one care home and two schools were not delivering enough public benefit.
Further public benefit assessments later next year will look at charities for the advancement of health and those working in sports or recreation.