It's misleading (especially to all the electronics novices around here) to say that a linear regulator does not increase current draw (in fact it does, slightly) when what it really wastes is POWER. For a testable, common example, let's use your normal Adventure. If you're powering the system from a 5V source, the power drain is 5 volts X 0.272 amps = 1.36 watts. Now using your 9 volt source, and assuming that the current really is the same, 9 volts x 0.272 amps = 2.448 watts input, an 80% increase. Or to put it another way, runtime would be 1.8 times as much if the regulator didn't waste all that power.
There is no perfect solution to this problem, but switching regulators are much better than linear regulators. Most are in the 80-85% efficiency range, with some approaching 95%. Here's an example calculation using a 9 volt battery with an 80% efficient switching regulator:
Take the desired output (5 volts X 0.272 amps = 1.36 watts) and divide by the efficiency, 1.36 watts / 0.8 = 1.7 watts input to get 1.36 watts out. Comparing this to the linear regulator, runtime will be 44% longer. Using a more efficient regulator circuit can increase this significantly.
This does not even take into account the ~2 volt "dropout" of the linear regulator. Using a "low dropout" linear regulator will increase runtime a bit, but a low dropout switching regulator will improve on that even more.