According to Fowler's Dictionary of Modern English Usage, which is a standard grammar reference book, the compounds that derive from the word "any" such as "anyone", "any way", "anyhow", "anymore" or the rare, "anywise" are all singular.
According to the Oxford Learner's Dictionary, here:
"any" can be used with uncountable or plural nouns in questions
The Oxford Learner's Dictionary does recognize that "any" can be used with singular nouns when it does not matter which one, such as in:
Pick any card.
The website referred to in the query is correct.
There is a difference of meaning in the two examples stated in the query.
The first sentence,
If you have any questions, please let me know.
is something a teacher would ask a group of students in order to make sure an explanation is clear.
The second sentence,
If you have any question, please let me know.
would mean the same as the previous example if "a" was substituted for "any". But as the sentence stands, it is a little unusual. It could be understood to be regarding the questioning of a person's guilt or innocence. "Doubt" would be a better word instead of "question" in this context.
The word "question" in the second sentence could be understood to mean "controversy", but this would also not be the usual construction.