Is dairy inflammatory ?
The short answer is NO.
The long answer is yes and no. For most people dairy is not inflammatory, for some people it is. If you are A1 milk intolerant (the common black and white cow), your body will react badly and dairy from A1 milk will cause inflammations in your body.
If you are intolerant to A1 milk, you are likely to be OK with A2 milk. This is milk from Guernsey, Jersey cows, buffalo, many other coloured cows, sheep and goat.
So if you are intolerant use buffalo mozzarella on your pizza (much nicer anyway) and cheeses and creams from any other A2 milk.