If there are only two airlines that fly between Dallas and New Orleans, what will happen in the market for one airline if the other one goes out of business?

If one of the two airlines, let's say, goes bankrupt and gets out of the market, then the other airline becomes a monopoly in this segment of the market. It is good news for the remaning carrier: it will face no competition, it'll be able to increase fares, it will dictate the conditions (reimbursement and cancellation policies) to its benefit. This is very bad news for customers, as they will have to pay more money, will find harder to make claims and will have only one option, at least for some time.