Is there any halachik problem with eating fish and milk together? (A classic example is a bagel with lox and cream cheese)
Famous question. Rabbi Yosef Karo in his commentary Bait Yosef (which would serve as a first draft of sorts of the Shulchan Aruch) writes not to eat fish with milk (Yoreh De'ah 87:3). His Ashkenazic counterpart Rabbi Moshe Isserles famously commented, "rabbi Karo mixed his milk with meat!" (נראה שנתערב לרב בית יוסף בשר בחלב; Darkhei Moshe, ibid), as the Talmudic quote regards fish with MEAT as being a health risk.
The fish-and-milk business was omitted from Rabbi Karo's later work, the Shulchan Aruch. So it's a question of how seriously we take the original comment in the Bait Yosef.
Non-Hassidic Ashkenazim simply follow Rabbi Isserles, that the whole statement was a mistake, and therefore there's no problem.
Among Sephardic Jews (who follow Rabbi Karo), and Hassidic Jews (who occasionally follow Rabbi Karo's stricter points), it varies from community to community. The mainstream Israeli "Eida Charedis" hechsher, which includes many Hassidic groups, will allow fish cooked in butter, but the Belz Hassidic hechsher won't. The Bukharian Sephardic practice may not be the Iraqi one, so you'd have to consult with your local hacham. Some rabbis also draw other differentiations, such as it's okay if they weren't cooked together (so yes bagel and lox, no tuna-noodle-casserole).
The Lubavitcher Rebbe writes in Reshimos: "The prohibition of eating Milk and Fish [mentioned] in the Bais Yosef is considered to be a mis-write. Nonetheless, we are careful [not to], therefore we add something (butter). This is an instruction from the Tzemach Tzedek". The editors of Shaarei Halacha Uminhag write that it seems that adding butter to milk makes the danger go away.
To continue Shalom's answer, I would like to bring down the opinion of the Taz that says the Bet Yosef made a misprint and meant to write fish with meat. So wrote the Hida. However, Maran HaRab Obadia Yosef in Yechawe Daat writes "Yesh Lehimana" meaning that is prohibited according to Sephardim. Rab Obadia Yosef brings a Kula from the Kaf HaHaim that butter with fish one can be lenient. For Ashkenazim it is Mutar.
There is a mishnah that explicitly permits fish and milk:
כל הבשר אסור לבשל בחלב, חוץ מבשר דגים וחגבים
My translation: "One is not permitted to cook [and eat] any meat in milk, except for the meat of fish and [permitted types of] locust"
So no, there is no halachic problem with eating fish and milk together, or else the Mishnah, our edition of Torah SheBe'al Peh, would not have permitted it.
Some forbid it for medical reasons, as per the Bet Yosef:
בית יוסף יורה דעה סימן פז
ומכל מקום אין לאכול דגים בחלב מפני הסכנה כמו שנתבאר בספר אורח חיים סימן קע"ג
My translation: "Nevertheless, one should not eat fish in milk because of its danger, as will be clear in volume Orach Chaim, Chapter 173 (where he does not mention fish and milk at all).
As mentioned in the answers above, most dismiss this as a misprint.
to all of the above learned commentors; you are all correct but not entirely. Not all ashkenazim are lenient and not all sefardim are stringent. yes, it is based on a beis yosef and whether or not he mixed up the words meat and milk, but their were even rishonim who were machmir, including rabbeinu bachaye! some feel its an actual medical issue and not the same as tzaraas mentioned by mixing meat and fish. Moroccans are known to be lenient entirely, other sefardim bdieved; the ben ish hai was extremely stringent. and although most ashkenazim are lenient some are mkpid due to the levish, pri megadim and g.maharsha being makpid for it.
you can read about this an a thorough 2 part article here