[I thought about that question as it became even more relevant with the advent of cooking TV shows where a Jew might be involved in cooking non-kosher foods with no intent to eat it or serve it to Jews.]
I see two major and two minor issues which, practically, would make participation in a cooking school complicated (there might be more - would love to get feedback).
- the prohibition to cook (and not just eat) a mixture of meat (from a kosher animal) and milk
- marit ayin, giving the appearance of doing wrong
- an additional issue on Pesach since a Jew is forbidden to own hametz - so any hametz he would be given to cook would be problematic
- similarly orlah fruit in Israel would be a concern (see here on 3 and 4)
There might be ways to circumvent these prohibitions, e.g., cook only poultry or meat from non-kosher animals, or cook in non-kosher milk (SA YD 87:3). Alternatively ohr.edu suggests a non-Jew could light the fire while a Jew cooks.
Interestingly, Rav Moshe Soloveitchik ruled that one may cook meat and milk together for a scientific experiment in which there is clearly no intention to eat the mixture (see Rabbi Hershel Schachter's "Peninei Ha-Rav," p. 152, quoted here). A rav would have to be consulted to see if one can extend it to cooking in a school since there is a benefit as well (learning to cook) but no intention for Jews to eat it.
2. Marit ayin prevents a Jew from doing anything that gives the appearance of transgressing the law, even if he technically doesn't. A relevant example here is eating meat in almond milk.
R David Brofsky writes there is a machloket poskim (rabbinical dispute) whether marit ayin applies in such a case. The Rashba (Teshuvot 3:257), and later the Rema (YD 87:4) rule that a priori one should not cook a non-kosher animal in milk because of mar'it ayin. The Shakh and Taz both challenge this ruling.
For a Jew to cook in a cooking school, he would have to explain all the precautions he is taking not to transgress halacha, incl. the fact he wouldn't eat the food or serve it to Jews.
As always, speak to your rav before attempting anything you have read about on the Internet.