Those would still be grammatically valid sentences with the omissions of "any", but "any" provides some particular emphasis, so the revised sentences would not necessarily have the exact same meaning.
If philosophers were made presidents instead of politicians, there would not be any war.
In the first example, the sentence without "any" might be understood as "there would not be a war as a direct consequence of the proposed situation." However, the sentence with "any" clearly expresses that no more war will exist at all.
The second example would likely have the same understood meaning with or without "any". However, "any" is used to strongly emphasize the dire situation that a complete lack of drinking water would be.