The NFL is no longer tax-exempt and the ownrs/teams never were.
the NFL is no longer a non profit. They likely still get big tax breaks as many companies get varying tax breaks, write offs etc..
However, the NFL still has an Anti-Trust exemption that allows them to fix prices, fix wages, and illegally centrally control media contracts, all of which are otherwise illegal. The teams also have property tax exemptions and the agencies financing their stadiums are floating tax-free bonds. This is worth more than all the other tax breaks and non-profit status ever totaled.