There are two ways in which not using reverse will save fuel. First is the obvious, in that when the engines are throttled up to fairly high RPM for the reverse thrust, that burns significantly more fuel than leaving them at idle. So forgoing reverse thrust saves that fuel burn.
Beyond that, engines can be shut down after landing after they've had whatever cooling interval is recommended by the manufacturer, and in some cases, the power setting on approach is low enough that this time "counts" as cooling time, so if reverse thrust isn't used, then an engine (in the case of a twin) or a pair of engines (in the case of a 4-engine aircraft) can be shut down shortly after clearing the runway. There is fuel savings here as well, not having to run the engine/s for the minutes required to give them the recommended cooling.
Additionally, not using reverse thrust can save on noise, which can be a "good neighbor" policy at urban airports at night. Some airports & carriers will be more concerned about this than others.
Depending on where the aircraft will park, it may be advantageous to use the entire length of the runway (i.e. if you're landing to the west and you will park at the west end of the airport), and for a really long runway with ample room to slow down without reverse thrust or heavy braking, it can make sense to "coast" out to the end of the runway.
Three reasons that using reverse can be worthwhile:
- Using more reverse means using the wheel brakes less. They may be more expensive than the fuel & wear on the engines from using reverse thrust.
- On a slick runway, the brakes may be ineffective at high speed, and the reverse thrust may be necessary to stop in the available runway. Some aircraft are certified in such a way that they have to be able to stop without the use of reverse thrust, so getting the TR's out is simply "bonus" stopping margin. Others, however, are allowed to take credit for reverse thrust in the stopping calculations, so for those aircraft on sufficiently slick/short runways, using the TR's is vital to stopping within the available runway.
- Finally, some operators require the pilots to use reverse thrust on every landing, even when it isn't needed. This is to preserve the habit patterns & muscle memory so that they deploy the reversers every time, not just "when needed." The logic here is that if you're making a decision at touchdown ("did I need them this landing?"), you may make the wrong decision when you really do need them, and the consequences in that case are very bad. So rather than risk a bad decision, the airline policy is, don't make that decision -- always deploy them.