Actually do is out of place here.
A. It is important to not have ...
= "Es importante no tener ..."
B. It is important that we not have ...
= "Es importante que no tengamos ..."
C. It is important that we do not have ...
= "Es importante que no tenemos ..."
A (with the infinitive) ~ B (with subjunctive), but C (with indicative) has very a different meaning.
I think that the sentence A has a very general meaning: it would be like to say that
in general something is important.
On the other hand, it seems to me that in the sentence B the verb stretches out into the future: I would use it to tell that
in order to reach an aim it's important that something has or not something else...
The sentence C would indicate a quality of something, thus the verb in this case would strech into the past: I would use it to tell that it's important that something has or not a certain feature.
For example, if I say "it's important that we not have troubles" (sentence B), I mean that if we not have troubles we can reach our aim.
If I say "it's important that we don't have troubles" (sentence C), I mean that it's good that so far we have not had troubles.
Finally, if I say "it's important to not have troubles" (sentence A), I would mean that, in general, it's good to not have troubles, and this sentence could be employed in both former situations.
I'm also studying spanish and I believe that the same goes for it. But this line of reasoning it's something mine, and I'm note sure of it to be correct. Is it?