Why would it? Looking around it looks like fuel makes up 30% of airline's expenditures. Disregarding profits, that means $300 of a $1000 plane ticket is for fuel. The current world wide price for jet fuel is $3.76/gal, and burning a gallon of jet fuel produces 9.57kg of co2. One source says carbon needs to be priced at least $100/tonne[3], which is in line with current prices in the EU. With that price, the price of carbon is expected to add $0.96 per gallon, or a 26% increase in fuel costs. However, it would only raise overall expenditures by 7.7%. I doubt that's doubt that's going to bankrupt airlines. Consumers will probably fully absorb the cost.
[1] https://www.iata.org/en/publications/economics/fuel-monitor/
[2] https://impactful.ninja/the-carbon-footprint-of-aviation-jet...
[3] https://www.reuters.com/business/cop/carbon-needs-cost-least...