While there was an ancient custom of fasting on the day of a yahrtzeit, it’s seldom observed today, and Chabad doesn’t adhere to this practice.
Those who fast don’t do so if the yahrtzeit falls out on Shabbat or a day when Tachanun is not recited. Special occasions, like a son's circumcision or a bride and groom’s Sheva Brachot, also override this fast.
If fasting, add the “Aneinu” paragraph in the Amidah (in the blessing of Shema Koleinu).