Is a public hot water urn, used by many people for both dairy and meat products (eg. Coffee with milk, meat instant soup mix), problematic? Assuming no non-Kosher products were used, and the only issue is Basar V'Chalav.
Now we have a very simple idea over here, okay. See the Tur in the very end of siman צב (that would be in Yoreh Deah) who quotes the Rosh saying that in a situation with two pots, one meat and the other milk and they are on top of one another, the steam from the bottom pot can become absorbed into the top pot thereby treifing up that pot. There is a machlokes achronim in how to understand this Rosh. The Bach (teshuvos hachdashos siman 21?, maybe 26? check it out for yourself, he talks about this issue as well as basar shnisalem min ha'ayin), holds that the situation is limited to when the pots are in a very concealed area (for instance, that lip of the bottom pot is sealed to the bottom of the top pot), only in this situation will the steam be able to asur the pot. The Trumas Hadeshen (psakim uksavim siman 103) holds that the factor making it asur is not the distance between the two pots, rather it is the tempurature of the steam as it touches the other pot. Both the Bach and the T"H (Trumas Hadeshen) use these explanations to matir a certain minhag (ע"ש). Now, that means that the main difference between these two shitos is when the steam is hot but not in a concealed area.
Now, back to the case in question. Okay. According to the T"H, the steam coming up from the cup of coffee may be yad soledes bo (very, very hot, hide your babies, ע"ש) and therefore the dairy-ness goes into the spout (and if you say that the steam is not hot enough at that point, if the spout is hot enough it also asurs it according to many poskim, see R' Akiva eiger on shulchan aruch, the pri megadim there, the beis yosef, the Toras Chatas and the levush) once someone else uses the pot for their meaty soup mix, the flavor may go into the soup from the pot. However, according to the Bach, the pashtus is that any area where you would be making this coffee would be too open of an area. okay. However, see the aruch hashulchan in (Y"D 92:55-56) who asks an amazing question on the T"H, which is that the Rosh himself in the teshuva says that if the upper pot is full of hot water then it would be too hot to accept to bliyos (absorbtions) and the upper pot would remain mutar, yet that is davka where the T"H asurs! And you should really see the T"H inside because he admits on his own that he didn't have this Rosh, he only knew about it from the question, and this makes the Aruch Hashulchan's question quite a bit stronger. Not to mention that the Bach himself says that we should be machmir for the T"H, ע"ש!!!
But we're not done yet, not by a longshot, okay. Because, as i mentioned (albeit in not the clearest way) many many very choshuv poskim hold like this T"H. TBH i don't know what to do with that information. See the Chidushei Hagahos on the Beis Yosef הנ"ל.
Another tzad lehakel could be that the kli containing this hot water may not be a kli rishon even if there is a heating element inside. I once asked a very prominent Rosh Yeshiva in E"Y about this and he answered based off the rather famous Tos' in shabbos (40b) that if the walls of the pot are not the medium through which the water is being heated then it is not a kli rishon but rather a kli sheini (the water might still be kli rishon water given that the heating element is directly heating it - as is the case in public hot water urns which are expected to keep a lot of water hot for a long time and to quickly heat up incoming water, but the walls which are being heated by the water may only have the status of a kli sheini) and kli sheini does not take in bliyos (absorbtions).
However, this may be difficult to rely on lechatchila given that the mechaber/rama in siman 104(?) say that we are noheg to be choshesh for bliyos even in a kli sheini lechatchilah. Furthermore, this chidush of the Rosh Yeshiva seems to be contradicted lehedya in shulchan aruch, who pasken that a kli that was left inside kli rishon water attains that status of a kli rishon (nafka minah, serving soup with a metal ladel, ודו"ק), while we says just now that it does not attain such a status. Now, I asked this exact question to this Rosh Yeshiva and he responded al regel achas that you could be mechalek, okay, because in the case of the ladel, the kli is fully submerged in the hot water, whereas the walls of this urn are not fully submerged but only touch the water on one side. (discuss the svara for this amongst yourselves. okay.)
Another potential issue to worry about in terms of the transfer of ta'am (flavor) into the upper pot would be nitzok, now, that means a stream of flowing liquid connecting two bodies of liquid. Although, so far as I have seen, pretty much every major posek in yoreh de'ah says that if the upper source is hot then it cannot recieve ta'am from the lower source. I apologize but I forget which siman these poskim can be found in.
Assuming that flavor could in fact become absorbed, like the majority of poskim would seem to say, and assuming that both meat and milk flavor could get in there, there would probably be no more that an issur derabanan of bishul basar bechalav (it would be derabanan if anything at all because it's not derech bishul, see the Pri megadim's hakdama to hilchos basar bechalav right before siman 87).
Regarding whether the flavor would ever become pagum (nasty, and therby having no negative ramifications in kashrus, ואכמ"ל), I recall seing a Shach who says that if the ta'am is ta'am heter, then even if there is a heat source reapplied to the ta'am within 24 hours it will still become pagum, okay, whereas if it is ta'am isur (like basar b'chalav) then it would continue to be good (dehaynu, not disgusting. it's still asur though) flavor until 24 hours pass without any reawakening of the flavor through heat.
Rabbi Ribiat has a little on this topic in Migdal Dovid footnotes 320 and onwards for a bit in meleches bishul, ע"ש and in the english part of the sefer where he is extremely machmir.
However, given the Aruch Hashulchan's kashya on the T"H, it would seem to be a pliyah atzumah (great flabergastation) to be machmir to such an extent. I've asked a number of very prominent Rabbanim and poskim in E"Y who all have an unbelievable mesorah in psak and they were all matir lechatchila (one told me in his characteristic way "there's no such thing as ze'ah (steam)!". Another told me in his own very characteristic way "Now, there's no ze'ah unless it's in a very enclosed space like an oven. Whoever told you that [there is a reason to be concerned for an isur] doesn't know the aleph beis of Yoreh Deah!")
Now, the Torah is not as pashut as people want to think it is, okay. Sof kol sof, I hope this was helpful, but you should still ask your local orthodox Rabbi, assuming he knows his aleph beis.