Can you take a Challa knife to any knife sharpener or would you have to Kasher it after it is sharpened? Would there be a difference if it is a meat or cheese knife?
Rabbi Forst's "The Laws of Kashrus" references a Mordechai cited at the end of Beis Yosef 122 which permits using a single knife sharpener for meat and dairy, though he does note that they must be clean and that some have the custom of using different sharpeners (page 354).
I was unable to locate the precise reference inside the Beis Yosef, but the context seems like it is probably speaking about using a non-Jews knife sharpener.
Indeed the Star-K specifically allows using a knife sharpener that is used for non-Kosher knifes providing that the knife is clean. Both the Star-K and the CRC Article on Knife Sharpening speak of the issue of the heat generated by electric sharpeners used by sharpening services and both conclude that the heat is generally not sufficient to effect a transfer of taste (this would seem to implicitly support my gut feeling that even though with respect to knives the pressure can effect a transfer of taste even without heat, that characteristic of knives isn't relevant here). They both note that one must be able to ensure that the knife they are getting back is their own.