Sorry, No Airline Ticket Tax Refund
I hate to be the one to break it to you, but you won't be getting an airline tax refund. No one will.
It's like the government and airlines are shouting a collective "Psyche!" First, it appeared that airline passengers might get a tax holiday on tickets purchased after Congress adjourned July 22 without passing legislation authorizing the Federal Aviation Administration's operating authority, leaving the administration without the ability to impose airline taxes. But most airlines decided not to pass on savings to travelers. Instead, they raised base fares to offset tax savings (see Airline Travelers Not Getting a Tax Holiday).
Then it seemed that people who bought airline tickets on or before July 22 for travel on or after July 23 were due a refund of the federal taxes they paid on their tickets (see Want an Airline Tax Refund? Be Patient). But Congress reinstated the FAA's operating authority today and, according to the IRS, reinstated airline ticket taxes retroactively. So passengers won't get a refund, after all.
On the bright side, "the IRS intends to provide relief for passengers and airlines with respect to ticket taxes that were not paid or collected because of the lapse," according to an August 5 IRS statement. What that means is that the IRS won't be collecting back taxes from those who purchased airline tickets during the tax lapse.
The IRS expects airlines to resume collecting ticket taxes by 12:01 a.m. August 8.