Learn French: Je dois serrer toutes les vis. - I have to tighten all the screws.

No, it is a notable exception! The 's' at the end of vis (screw) is pronounced, so it sounds like "veess". Notice also that it is a feminine noun, which is why we use the feminine plural toutes les.

How would I use the grammar from this lesson to tell someone they can tighten "any of them"?

Since vis is feminine plural, you would use the feminine plural pronoun n'importe lesquelles. For example, if someone asked which screws to start with, you could say Serre n'importe lesquelles (Tighten any of them).

What are some other common uses of the verb serrer?

In this sentence, it means "to tighten", but serrer generally means to squeeze, grip, or hold closely. You will frequently hear it in expressions like serrer la main (to shake hands) and serrer dans ses bras (to hug or hold in one's arms).

Why do we need les after toutes here?

When expressing "all the [noun]" or "all of the [noun]" in French, you must include the definite article (le, la, or les) between the correct form of tout and the noun. Without the article, toute would mean something closer to "every", though chaque is more commonly used for that.