In order to answer your question, we must address two different scenarios.
If you are the first to use the sharpener, then a simple precaution will prevent problems from arising. As long as you make sure that the knife’s blade is clean when sharpening, there will be no transfer of ta’am (absorbed taste) from the knife to the sharpener, which will then remain pareve. This is because ta’am is not transferred from one kli (utensil) to another without a capable medium.[1]
Before continuing, we must note that one should not take the cleanliness of a knife for granted, even when it appears to be clean. The gemara[2] says that a radish that was cut with a knife that is used to cut meat is assumed to be fleishig. Rashi[3] explains (as one of two explanations) that a knife often has a thin layer of fat on it, and this layer may come off onto the radish when it is cut. One might thus claim that the standard fleishig knife has residue on its surface, which can be transferred to the knife sharpener with the help of the friction and heat that are produced during the sharpening process. However, poskim assume that it is possible to clean a knife so that no residue remains. Only when one does not clean it right away, but rather allows the fat to congeal on the knife, is it insufficient to merely wash it clean. In that situation, ne’itza b’karka (plunging the knife repeatedly into firm earth) becomes necessary in order to properly clean it.[4] (Many say that using an abrasive cleaning agent, such as a strong detergent or steelwool, is equivalent.)
If the sharpener was previously used without care having been taken to ensure that the knives were clean, a problem arises. During sharpening, surface meat or milk residue may have gotten onto and/or into the metal of the sharpener. Even so, if you subsequently clean the sharpener and the knives that are to be sharpened before the next use, there will not be any transfer of ta’am, as we explained above that this will happen only with the help of some medium. However, if either the sharpener or subsequently sharpened knives remain soiled, the transfer of ta’am is possible. For example, if the sharpener remains soiled on its surface by milchig residue, a taste may be transferred into the blade of a fleishig knife that is being sharpened and, in addition, the mixture of the tastes will transfer back to the sharpener. If the sharpener is cleaned after being made milchig but another knife being sharpened has food on its surface, the milchig ta’am can be transferred from the sharpener to the food and, simultaneously or subsequently, to the knife. It is true that if the clean sharpener sat unused for 24 hours, ta’am that emanates from it is assumed to give off a detrmiental taste and the matter is far less problematic (details are beyond our present scope). However, this factor is insufficient to allow unrestricted, continual use.[5]
The question is if we can be optimistic about how the sharpening will be done or if we need to assume the worst. Hagalat Keilim (Cohen)[6] cites an opinion that does not permit giving knives to a non-Jewish professional to sharpen out of concern that the sharpeners will have a residue from non-kosher meat or fats. However, he points out that in our times, people usually give only clean knives to be sharpened. Furthermore, he implies that even during the time of the Rishonim, indications are that few poskim were concerned about this problem. For example, in discussing the issue of giving a knife to a non-Jew for sharpening, the Mordechai[7] raises only the concern that the tradesman will make personal use of it for non-kosher food, not that the sharpening itself is problematic. Rav Cohen concludes that it is proper and sufficient to carefully clean a sharpener before using it.
In conclusion, while it is conceivable that using one knife sharpener for both milchig and fleishig could cause problems, it does not seem necessary to be concerned about this possibility. The key is to take note that the knives that will be sharpened are clean and that the sharpener itself appears clean.