Are airlines in the US legally obligated to refund any airport tax charged if a seat was not used?
In particular I am thinking of a flight with Spirit Airlines from Fort Lauderdale to Boston which was missed.
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No (basically, although you might get a couple of bucks if you jump through enough hoops).
The terms of your travel are governed by the airline's contract. Some US airlines have language specifically addressing this case. For instance, American Airlines says on their website:
Upon request, taxes and fees not imposed by the airline may be refunded if the airline's obligation to remit the tax or fee depends upon use of the ticket for travel. Taxes and fees will not be refunded if the airline's obligation to remit the tax or fee arises from the collection of the tax or fee. All taxes and fees imposed by the United States government fall within this category and are not refundable.
Similarly, this USA Today article explains that the 7.5% federal excise tax and the per-segment tax aren't refundable "because the taxes are applied when the ticket is bought rather than when the flight is taken." The TSA fee, which is only a couple of bucks, could conceivably be refundable according to that article. You're welcome to ask Spirit for it, though knowing Spirit's reputation, I wish you lots of luck. Here's a 2012 article about someone who got a whopping $25 out of Spirit in taxes, but he had to get a newspaper columnist involved before they'd pay.
That said, if you have a refundable ticket (which Spirit does not offer), airport charges and taxes can be refunded along with your airfare if you qualify for a refund. And if you change your reservation (after paying any applicable change fees), the taxes can be included in the total value of your ticket when your new itinerary is repriced.