Does this sentence have two meanings?
Don't forget always to retighten the screws.
In this situation I want to tell the listener that he should always retighten the screws.
I do not want to tell him that he always forgets to do that!
Does this mean I should say:
Don't forget to always retighten the screws.
I know it's okay to split the infinitive (especially in AmE), but should I do it with always?
On a related note, the term "split infinitive" is a misnomer. You're not splitting anything because "to" is not part of the infinitive; it's a preposition.
See To split or not to split? | The Grammarphobia Blog.
Although there are innumerable precedents and arguments in favour of split infinitives, many people still try to avoid them. I'd just drop the 'Don't forget'. Or if it's important to keep those words, you could still avoid the infinitive, by adding a colon: 'Don't forget: always tighten the screws.'