APU usage, over many flights and a longer period of time, will certainly impact on the company bottom line. That is why companies will have standard operating procedures dictating when to turn on the APU during preflight to minimise fuel costs (around 30 minutes before departure typically).
However, in terms of fuel planning for a flight, APU usage is not really an issue. Depending on aircraft, a typical APU on a heavy will run somewhere in the region of 600-800kgs of fuel per hour. Not inconsequential (for example, it will be an issue if you have to run it inflight), but then again, not really too much of a planning issue either. At some airports, we waste more fuel taxying than having the APU on for an hour.
And as Lufthansa mentioned above, our primary concern is the welfare of the passengers. If ground air is not available, APU will most certainly go on, regardless of circumstance. And ground air is not always available.
Generally speaking though, we do save fuel in a variety of ways. Cost index is the main one, as the aircraft will generally fly a more economical speed with a lower cost index, taking into account things such as specific fuel consumption, and specific ground range.
Higher levels will also help, as turbine engines run more efficiently at higher altitudes (the higher the engine rpm's, the more efficient they are generally, though there is a practical limit.)
Shortcuts, or directs as we call them, will save a bit of fuel and time, though that is very much an ATC constraint. Will get it mostly in North America, Japan and Europe, but less flexible in China or Russia.
Reducing bleed usage is another method, though now we're going into small details. For example, anti-ice usage is normally at a minimum, packs are run on low if there are less passengers, etc.
Judicious checks of the aircraft C of G will impact upon economy also, as on the larger jets, this can be controlled via piping fuel to the aft trim tanks on the horizontal stablizer. The operation is automatic for the most part, but still important to make sure that it is operating normally, with large fuel loads on long haul flights.
As to reduced power takeoffs, they're not really for saving fuel. Reducing the power for takeoff can reduce engine wear, but increases noise (which is why LHR
requires max takeoff thrust in order to clear the noise abatement areas at a higher altitude) and fuel burn because of a slower climb to cruise.
As to some of the other ones, Don has already mentioned a few.