Why don’t airline ticket prices change when the price of gas drops?
It's supply and demand.
Airlines purchase large amounts of fuel from refineries and/or wholesalers and distributors

Fuel is a major expense for airlines and is a factor in airline ticket costs. Airline operators must plan in advance to secure supplies of gasoline, diesel, and jet fuel, and typically buy at market rates from refiners or from wholesalers/distributors. Airlines use a number of hedging strategies to control their fuel costs.

But, there is a lot more to airline ticket prices than meets the eye

The price of your ticket consists of a number of things, including: the base fare, taxes and airport fees, fuel surcharges (if applicable), service fees to the issuer, food, seat selection, baggage, and more depending on your carrier. Airlines offer a number of travel classes (e.g. First or Economy) as well as a variety of booking classes. Booking classes are different types of tickets within each travel class that have different rules, restrictions, and prices. These booking classes appeal to different types of travelers and help airlines maximize profits. 

Prices change due to seat availability and demand
Prices can vary widely, even during the same week. Airline competition also plays a large role in ticket prices. If there are a lot of airlines flying to a particular destination, prices will be significantly lower. This applies to both certain regions and certain routes.
While fuel prices do matter, they are certainly not the only factor in airline ticket prices
Airlines use dynamic pricing strategies to adjust fares to account for their costs, past bookings, remaining capacity, demand, the probability of selling more seats later, and competition from other airlines. Airlines use sophisticated technologies and algorithms to determine the price that you see online.
Did you know?
Average round-trip US domestic airfares have dropped more than 40% since 1978 (http://airlines.org/accessible-age/).