I doubt it was millions of taxpayers dollars, but it was certainly a large amount.
More than that, he deceived and horrified an entire nation. I'm more offended by the millions of people who were played for suckers than by the money spent - as bad as that also was. And yes, whatever the total comes to, assuming he's found gulity (and the sheriff talks like the evidence is compelling), he should be on the hook to repay it.
There are state and federal restitution laws. The FAA can charge him for re-routing airplanes from DIA. There are the choppers from the Colorado Air National Guard which costs about 5 thousand dollars an hour.
They can charge him for the amount of man hours it took to search for the little boy in several juristictions.