Someone told me that there is a Minhag (by some) not to eat dairy the entire Pesach. I was wondering what the reason is?
I have heard that there are those that do not purchase dairy on Chol Hamoed for the reasons mentioned below. I have never heard of people that do not eat any dairy at all over Pesach, however maybe there are some that do not.
Technically speaking, there is no concern with Chometz when it comes to milk. Cows eat grain, which becomes Chometz with prolonged contact with liquid. The custom to purchase dairy products before Pesach is rooted in the fear that a Chometz particle may be present in the milk.
How does it help if you purchased your milk before Pesach? The answer is Bittul. A particle of grain would be indiscernible and also would constitute less than one-sixtieth of the milk. Therefore the grain is nullified in the milk. However, nullification applies only before Pesach begins. On Pesach we do not apply the rule of Bittul. Therefore, if the milk was in the possession of the Jew before noon on erev Pesach, any possible Chometz in the milk is nullified; hence the custom to purchase milk before the holiday.
I heard that the reason for the custom was that one is not allowed to drink milk that was milked over pesach. Before refrigeration, that leaves only the first two days one can drink milk. Given that one should meat on Yom Tov, that doesn't leave much time to drink milk. Moreover, to buy new dishes for two days(like mentioned in an answer by JXG) was expensive (they had no disposable dishes). Therefore, people didn't drink milk.
By Pesach, the custom is to be extra stringent in Minhagim, and to avoid anything one's ancestors didn't eat, even if the reason why they did not eat it doesn't apply anymore (For example, Gebroks, Kitnyos, etc.)
I know of families whose practice in Poland was not to cook dairy on Pesach, because they couldn't afford the extra set of dishes. They drank milk, though. Nowadays, I haven't heard of it.
In Iran, the milk in the city I'm from was all procured by non-jews. So unless you bought a whole cow for pesah, you wouldn't have dairy because there was no means of verifying if there was hametz in the milk.