A tax-exempt organization does not escape filing, it files differently. The annual return in the 990 series reports your finances, governance and program activity to the IRS. Which version you file depends on size: the full Form 990 for larger organizations, Form 990-EZ in the middle, and the 990-N electronic notice for the smallest.
Two things make this different from a business return. First, the 990 is a public document that donors, grant-makers and journalists can and do read, so how it presents your organization matters. Second, failing to file for three consecutive years costs you exempt status automatically, and getting it back means reapplying from the beginning.