Which sentences are correct?
- We didn't have air conditioning.
- We didn't have the air conditioning.
- We had no air conditioning.
Which sentences are correct?
1 or 3. There is no definite article because you are not talking about a specific air conditioner system
"Air conditioning" is a process, or abstract noun. Grammatically, it is similar to other abstract nouns such as "peace", "love", and "justice". Abstract nouns always take the zero article, as in your examples #1 and #3 (both of which are correct).
To change the abstract "air conditioning" noun into a concrete noun, you would say "air conditioner", which is the physical machine as opposed to the process of having the air be conditioned.
We don't have air conditioning. {zero article}
We don't have an air conditioner.