It depends on what you are referring to by zero dollar return.
If you simply have no income to report, then - no - a federal tax return would not be required to be filed.
If you have income, but you are just not receiving a refund, nor do you owe taxes (so a "zero" refund), then it is still possible that you may be required to file a tax return depending on your income.
Read these two articles below - the first one is for general taxpayers and the second is for dependents.
Note: While you may not be required to file a federal return, you may still need to file a state return as their requirements can differ from federal. You can view this article for more information about state filing requirements.