In order to prevent accidental mixing of meat and dairy, Chazal felt it necessary to enact additional restrictions including not using fleishig utensils with milchig food [and vice versa] even when everything is cold.  The general concern is there may be shamnunis (residue) that remained on the table or utensils that might rub off onto the food, resulting in someone inadvertently eating meat with dairy.  It is important to note, that since the concern is for some remaining residue, even a non ben-yomo (day-old residue) can potentially prohibit the food.

Therefore, one may not eat fleishig on the same table or tablecloth that milchig was eaten [or vice versa] even though one is finished eating the "opposite" food (Shulchan Aruch YD 89:4).  There is an opinion that this halachah only applies if food was placed directly onto the tablecloth.  However, if one uses a plate it is permissible to use the same tablecloth (Pischei Teshuva 89:8).  Many opinions argue and say that one may never use the same tablecloth even if one did not put the food directly on it (Darkei Teshuva 89:48).  One may use a very long tablecloth where he uses one side for fleishig and the other side for milchig, as long as he has marked off which side is for which [to prevent accidental use of the wrong side] (Darkei Teshuva 89:48 & Badei Hashulchan 89:101).

One is not obligated to have a tablecloth that is exclusively fleishig and exclusively milchig.  As long as one thoroughly wipes down the tablecloth, one may use it for the other (Magen Avraham 193).  Similarly, one is not required to have 2 separate highchair trays as long as the tray was wiped clean (Rabbi Biberfeld, shlit"a quoting Rabbi Shmuel Felder, shlit"a).  The reason is, generally one does not put down something that is hot enough [yad soledes bo] directly from a kli rishon, therefore the taste never absorbed into the tablecloth or the highchair tray.  In our society, one generally wipes a table and tablecloth clean between meals; therefore, we can assume that the table was clean (Laws of Kashrus pg. 211 ftnt. 107).  Nevertheless, the common custom is not to eat fleishig and milchig on the same tablecloth (Badei Hashulchan 89:102, Laws of Kashrus pg. 211).

It is advisable to have separate fleishig and milchig dish towels, since there is a concern there is food remaining on them or it touched hot food which came from a kli rishon (Kashrus in the Kitchen pg. 91).

It is prohibited to leave uncovered salt [e.g. in a salt dish or bag] next to open milk (Shulchan Aruch & Rama YD 95:5).  The concern is, the milk might spill in the salt and one may accidently use the milchig salt with fleishig (Taz YD 95:16 & Shach 95:22).  If one inadvertently left them next to each other, it is permitted to use that salt with fleishig, for one does not have to assume that it spilled (Rama YD 95:5).  The Poskim write that it is a minhag yafeh (praiseworthy custom) to have a separate salt dish for both fleishig and milchig (Rama 88:2 & Taz 95:18).  The reason is, people would dip their food or dirty hands directly into the salt and there might be some remnants of fleishig or milchig.  Accordingly, this reason would not apply to salt shakers, since one does not place the food directly into them.  Nevertheless, the common custom is to have separate saltshakers, for the concern that they might directly touch the food [especially in a house with young children] (Badei Hashulchan 88:30, Laws of Kashrus pg. 212).  Some Poskim suggest that in a household of young children, having separate saltshakers might be the strict letter of the law and not merely an added stringency (Badei Hashulchan 89:99).  This halachah would apply to any food or spice [e.g. spices, ketchup and mustard] that one uses with both fleishig and milchig, if there is a concern that they might directly touch the food (Badei Hashulchan 88:30, Laws of Kashrus pg. 213).