Mixing fish and meat together is not restricted by Biblical law.
However, the Rabbis of the Talmud considered it a harmful mixture, and they forbade it for health considerations. The Gemara Pesachim (76b) teaches that one may not eat fish and meat together since this combination is considered sakana (harmful). Still, the Gemara is not explicit about what particular health hazard (dabar acher)meat and fish would trigger.
Shulchan Aruch (Y.D. 116:2-3), following Rashi's opinion, identifies this danger as tsara'at, traditionally understood as a type of leprosy. By the way, and public opinion notwithstanding, Maimonides explicitly mentions that sara'at should not be identified exclusively as leprosy. He hints that sara'at was more a category of a (infectious?) disease (T. Tsarat 16, 10) than a particular disease.
Modern Rabbis also disagree about the specific health issue involved in mixing fish and meat.
"The combination of these two foods [fish and meat]can have negative health effects that are not readily apparent. Even if modern medicine does not recognize these health concerns, we can never be sure that the concerns are outdated" .
It could have been allergy or even, as I've heard from one of my teachers, the risk of swallowing a fish bone hidden in a piece of meat.
In any case, the prohibition of eating fish and meat together remains intact today.
The Rabbis sentenced "Chamira sakanta me-isura", when it gets to preserving one's health (sakana), we should be stricter and more careful than when we are dealing with a religious prohibition (isur).
Rabbi Yosef Bitton. YMJC 130 Steamboat Rd. Great Neck NY 11024