There is a time-honored practice to fast on a yahrtzeit - anniversary of a death - and especially on the yahrtzeit of the truly righteous. However, we do not find such a practice practiced by Chassidim. This follows the general philosophy of Chassidut that in later generations we should minimize fasts and self-punishment.
The Baal Shem Tov propounded this philosophy by explaining the verse: When you see a donkey...lying under its burden...it may occur to you to follow the path of mortification of the flesh...not in this approach...rather...you must aid it...purify the body, refine it, but do not break it by mortification the first Lubavitcher Rebbe explains in his book of Tanya that in modern times we should not undertake fasts even for the sake of repentance, rather we should increase our Tzedakah - giving charity.
This prevalent weakness of the physical nature of the generation is not something we chose; rather it is caused by Divine Providence. G-d made these generations more frail, to show us that the refinements of the world that had to be accomplished through mortification were already accomplished by previous generations; it is no longer necessary now.
In these generations we can accomplish the same without pain and with joy and good health. Chabad specifically teaches that our Divine service must penetrate the human being (not break it) - start with the intellect and then permeate the entire being. So, too, in the case of the yahrtzeit of a Tzaddik, today we can receive the spiritual revelations tied to the day through Divine service related to the body without fasting.
[I do not challenge other groups or customs which sanction fasting "for each river flows in its own way" and everyone must follow and climb his own trail, but this is the road of Chabad. This is especially so in light of the earlier explanation of the elevation of the soul attained on a yahrtzeit, as well as the elevation of those associated to the soul of the departed. In the life of the concerned individuals this will express itself in more Torah study, more charity and more good deeds, etc.]