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A question for the libertarians in here: If the FBI got a call from a flight instructor who said a Middle-Eastern man paid $2,000 cash for flight lessons, would that justify a subpoena of the customer's financial records? Would it justify a wiretap? Would it justify any governmental action whatsoever?

From the amount of information you've given I'd say no. Seems some people I hold in high esteem would agree with me.

The 4th Amendment to the Constitution of the United States of America:
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.

If he paid cash for the lessons then acted suspiciously (i.e. not wanting to learn how to land) then yes I'd say a background check is in order, especially if the person is not a citizen of the USA, but just because someone of middle-eastern decent pays cash for something that people in this country do everyday is not a cause to raise the alarm.
Someone paraphrased Ben Franklin's “Those that would sacrifice liberty for security deserve neither liberty nor security.” in an earlier post and I fully agree with his sentiments.