Although it would be perfectly natural to omit any in a very wide variety of utterances, there is the possibility that its absence might be understood as the speaker's way of hinting at some sort of special circumstances, as a leading statement that will have a follow-up:
I had my mobile phone with me but I didn't take pictures with it.
-- What did you do with it then?
I recorded some bird calls. I could hear them up in the trees but I couldn't see them.
whereas by contrast the same statement with any is more likely to be understood as a simple statement of fact and would not awaken in the listener any curiosity to know more:
I had my mobile phone with me but I didn't take any pictures with it.
That second statement is likely to be understood as "I don't have any pictures to show you." End of story.
So, any and the omission of any can have nuanced meaning in conversation, and there is no crude rule such as 'any' is a non-essential word that can be used or not used at will with no change of meaning.