According to Ashkenazi tradition, "If a meat/milk pot hasn't been used for a year+, it may be used as the other type without Kashering". If I have a ceramic coffee mug that may or may not have contained a hot non-kosher food/liquid more than one year ago, may I now use it?
The רמ"א as understood by the Magen Avraham (509:11) quotes a long-standing tradition not to kasher utensils from fleishig (Meat) to milchig (dairy) or vice-versa, because it will by mistake lead to mix-ups and halachic problems.
There are, however, certain leniencies regarding this Tradition:
The פרי מגדים (Orach Chaim, Eshel Avraham 509:30) says that if a person has an extra set of fleishig silveware that he wants to use for milchig, then he cannot simply switch it over to milchig. Instead, he can treif it up on purpose, and then kasher it and use it for milchig. Some find this suggestion problematic.
The חתם סופר (Shu”t Y.D.110) says that what one can do is kasher the fleishig silverware for פסח, and then use it for milchig during the year and it will avoid issues of confusion that the רמ"א was worried about.
The מהרש"ם (Shu"t Maharsham 2:241) says that if a whole year has passed since the dishes were last used, then one can kasher them from one to the other even לכתחילה. This is based on an idea that is found mainly in the חכם צבי. The halacha is that after 24 hours it is מותר to use a meat pot for dairy on the level of דאורייתא, but the דרבנן disallowed it. The חכם צבי says that this גזירה only applies through 12 months (he bases this on a גמרא regarding סתם יינם, which says that any barrel that was אסור due to having had סתם יינם inside it can be used after 12 months, because whatever טעם that was inside it has disappeared. He says that this proves to us that טעם disappears after 12 months, and the same holds true for the טעם of meat or dairy in כלים as well). The מהרש"ם says that even though we do not rely on the חכם צבי to use dishes after a year as though they have lost their old status, we will rely on him to kasher from milchig to fleishig after a year. Rabbi Sobolofky, Rosh Yeshiva in Yeshiva University, felt that one has the right to follow this מנהג of the מהרש"ם, to wait a year and then kasher לכתחילה.
It comes out from these sources that that since the whole reason for the tradition brought down in the רמ"א was it would cause the mixing up of dairy and meat dishes, so when dealing with a Non-Kosher dishes the פרי מגדים felt you are allowed to Kasher right away.
It must be noted that these rules are only true for dishes that are able to be Kashered. Ceramic (for all intents and purposes) cannot be Kashered.