I am in the US on an H-4 visa. Am I permitted to teach Bharatnatyam (classical Indian dance) classes to children in my apartment?
Working on H4 without EAD is illegal and puts you in violation of your status. It doesn't matter what kind of work it is or whether you get paid for it or not. What is forbidden without a valid work authorization is work, not earning money.
The problem with being in violation of your status is that it has a domino effect. Every step towards your green card or naturalization will require your certification that you have never violated your status. It only takes one disgruntled person to tip the USCIS off, and you'll be on your way home in no-time. Do you really want to risk it?
If you don't take any money, then you are not working -- you are just volunteering or having fun with the neighborhood children.
H-4 dependents are not permitted to work in the United States. H-4 dependents, however, may attend school on a full-time or part-time basis, without a new immigration application. They also are permitted to participate in volunteer work with nonprofit-type organizations, for which they receive no remuneration or benefit from the employer. If the work is simply an unpaid position with a for-profit type of employer, it may be considered unauthorized work and, thus, not permissible in H-4 status. This should be discussed with a qualified immigration attorney if questions arise. Source.
The intent of the law is that you should not be replacing any legal worker's wages. I can't imagine anyone seriously arguing that playing with kids in the rec room of your apartment building will fall under the rubric of an "unpaid position with a for-profit" or that you displaced a legal worker with your activities.
If you have expenses that need to be covered (clothing, equipment, music, etc.) then it might be best to ask one of the other parents to collect money on the parents' behalf and for these things out of that fund. You want to keep more than an arm's length away from this type (pr really any type) of financial activity, to be on the safe side.
That being said, the government really has better things to do than chase you down. As long as you make it clear you're that you're teaching on a voluntary basis, for fun, and not taking a salary, there shouldn't be any issue.
Note that this forum post deals with the OP's specific issue of dance classes. The consensus of the others is that volunteering without pay is acceptable.