Does current flow through the potentiometers of buck and boost converters?
Yes the potentiometers act as a variable voltage divider, so you can change the output voltage of the converter whenever you like. This voltage divider is connected to the Feedback (usually this pin is named as "Feedback") pin of converter IC, providing feedback for the regulation loop. 



The datasheet usually contains information about this pin's bias current. In case of the LM2596, it is typically 10nA.



The feedback resistors are also recommended in the \$ k\Omega \$ range, reducing current flow.