Although most Christian denominations do not require keeping kosher, are there any that prohibit keeping kosher? I'm most interested in Reformed, Catholic, and Eastern Orthodox views, but also others (to a lesser extent).
The Catholic Church does not forbid her faithful to follow any kind of dietary restrictions; the only restrictions she places on faithful are the fasting requirements during Lent — which, recently, are only obligatory in Ash Wednesday and Good Friday — and the abstinence of meat on Wednesdays and Fridays, or as directed by the country's Conference of Bishops. The details are in the Code of Canon Law, Canons 1250–1253.
The Orthodox Church also prescribes fasting on Great Lent, which is a period similar to Catholic Lent but starting from Clean Monday, which is the Monday before Ash Wednesday when Easter happens to coincide between East and West, and is similarly displaced relative to Ash Wednesday as much as Julian and Gregorian Easter Sundays are displaced when they don't coincide. In this period, Orthodox follow an essentially vegan diet, and are forbidden from wine and olive oil in a few of those days, in addition. Finally, Eastern Orthodox monks and nuns are on this Great Lenten diet permanently, as soon as they join the monastery.
Ethiopian Orthodox Tewahedo follow Jewish custom for ethnic reasons: they maintain they have Jewish ancestry from the time of King Solomon and the Queen of Sheba.
Finally, Reformed generally don't fast even in Lent, or if they do, they do so individually according to each person's conscience. The Heidelberg Catechism, for instance, makes no mention at all of fasting, diet or food (except for a section on the Lord's Supper).