"In any practical measurement of potentials, current flows in the measuring circuit for at least a fraction of the period of time over which the measurement is made. Ideally this current should be very small. However, in practical situations, it is never zero. Biopotential electrodes must therefore have the capability of conducting a current across the interface between the body and the electronic measuring circuit." - Michael Neuman in Medical Instrumentation [1]This is the case if the current which flows is used to measure the potential, for instance through a resistor. What if the potential is all that we are after? Do we need to still worry about facilitating this current? Is the potential meaningful without current flow? Or, can we measure potential without current flow?
We can accelerate electrons in vacuum from one metal plate to another by imposing a voltage across the two plates. In most instances of this experiment, the electrons are released from one of the plates, so current really is flowing. Assuming that it is possible to drop an electron into the the field, would it accelerate? It would, so based on the acceleration we could measure the potential across the plates. Perhaps an even simpler experiment would be to send an accelerated electron through a perpendicular field and use the deflection to measure the strength of the field. So, it is possible to measure potential without current.
Now if we think about it, there is some wierdness going on. Where does the energy come from to create this field which does real work on the electron? Isn't that strange? If this was the case we could deflect electrons all day long for free. I'm not understanding something here...
Reference
[1] M. R. Neuman, "Biopotential Electrodes," in Medical instrumentation : application and design, J. G. Webster and J. W. Clark, Eds., 3rd ed. New York: Wiley, 1998, pp. 183-232.