Ibrahim, and welcome to ELU.
First of all, you are right in your distinction between nouns that have no plural (like information) and nouns that have a plural (like reference). Strictly, nouns without a plural are called count nouns. It is not the nouns that can or cannot be counted, but the things to which they refer.
Some nouns can be used as either count nouns or as non-count nouns:-
Bread as in "Please pass the bread". In this sentence the only thing that can be counted is a loaf of bread, as in "Would you like two loaves of bread or one?"
Justice, as in "All people are entitled to justice.", has no plural.
But each of these words has a slightly different meaning, in which it refers to something which can be counted.
"Some bakers make many different breads (that is types ofbread); others specialise in just one (type of) bread."
This city's court operates with three resident justices (ie justices of the peace)."
Coming back to your main point, sentences like the one you cite are typical of how writers can find themselves in no-win situations. Whichever way round you put it, the sentence will jar. At least, it will jar on any native speaker of English paying attention to (some might say “fussy about”) what it being said.
Your second version is easier to bear because the plural verb is close to the plural noun. But this has nothing to do with count nouns themselves or with their opposites. You can generate the same discomfort without this.
Can you tell me whether six red roses or a single rose are/is needed?
There is a simple rule: avoid having more than one subject to a verb, if one is singular and the other plural.
And there is an easy way to do this in your case, by following another simple rule, this time one of style:
Prefer active verbs to passive ones.
So in this case, it is fair to assume that the people either in need or not in need of the information or references are those who commissioned the writer asking the question. If so, then write:-
Please let me know if you need any additional information or references.
Bang goes the problem.
Of course, there is another equally simple way round the problem in this case, which is to use the conjunction and in place of the disjunction or
Let me know if any additional information and references are needed.