There are countable and non-countable nouns. It sometimes sounds best to keep non-countable nouns with 'have no', while countable with 'do not have any.' This, I think is an aesthetic choice rather than a rule. We have no bread - He has no money - They have no class We do not have any spoons left - We do not have a ride today - We do not have any arrows in the quiver.

Correction: I don't have any lessons and I don't have a car. Here's a video explaining it to you http://www.engvid.com/english-grammar-do-does-did-dont-doesnt-didnt/