Airline margins are already razor thin, about 10%, so increasing costs by that much would make them very close to not profitable. Not profitable industries die
Air travel will get more carbon efficient. The whole idea behind a carbon tax is that we start low and hike gradually, giving the market time to respond (rather than flipping a switch and raising ticket prices 7% overnight). Of course, the longer we wait the faster we have to hike prices (the timeline compresses), so it's hard to feel bad for industries that have resisted carbon taxes for the last decades. Moreover, if climate science is correct, the consequences of doing nothing far outweigh the loss of cheap air travel (of course, that's just a worst-case scenario; we absolutely don't have to choose between those two outcomes).
You realize that companies can raise prices? That's wha I meant by "Consumers will probably fully absorb the cost" last comment. Supermarket margins are even thinner than 10%, yet they aren't going broke during this inflation spree.
Economics 101: raising prices reduces demand. If airlines raise prices less people will fly — demand for flights is fairly elastic — further eating margins. Grocery stores sell food, the ultimate inelastic item.