How unnatural is it to not use "any" with negative sentences?

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Both of your examples are fine with or without "any." There is a small difference in nuance but both versions are perfectly natural.

In a sentence, using "any" with a negative specifically means you have none at all. Without "any" this isn't necessarily the case.
Example:
#1. "I wanted to go, but I didn't have any time."
#2. "I wanted to go, but I didn't have time."
Sentence #2 means I didn't have *enough* time. Maybe I had no time at all (like sentence #1 says), but maybe I had some free time but not enough to go.

In a question, there isn't as strong a difference in nuance but I feel like there's still something. Perhaps the version with "any" feels more like the questioner is assuming an answer of "no" is more likely, while the version without "any" feels like the questioner's assumption leans toward an answer of "yes." This is a very subtle distinction that likely only matters when you're asking a favor of someone.
Example:
#1. "Do you have any time to help me this weekend?"
#2. "Do you have time to help me this weekend?"
Question #1 seems slightly more deferential, making it easier for the listener to answer "no."