The two are often interchanged, but there is a difference. By preceding time with the definite article 'the', you are changing 'time' from a general indefinite period to a time period that is more restricted or scheduled. In practice what this means is, "We don't have time" is closer to a general observation or statement, whereas "We don't have THE time" is drawing attention to the listener that you don't have an allocated period of time available to do it.

If the purpose here is to explain that there is not enough time available to complete a task, I would say they are the same. Either could be used. If there is any difference, I think you would more often hear someone say "we don't have time" about a task happening right now e.g.: "We can't go to the store now, we don't have time" (before it closes) Often "we don't have the time" is about a task that doesn't have a set timeframe: "We can't learn to cook, we don't have the time" (we are always busy)

We don't have time (at all) Time in general. We totally have no time. We don't have the time "The" means "this". The concrete period of time available now.