"Es" as a subject replacement (Expletivum) can generally be dropped from a sentence when it can be ensured otherwise that the verb is in the second (logical) position.
German, unlike many other languages (and that's probably why many non-native speakers feel that something must be missing), doesn't really need a subject to form a proper sentence, as long as the above condition is met.
In your example, the "doch" covers the requirement that the V2 rule is met by occupying the "Vorfeld". If it's removed from the sentence, it's no longer a grammatically well-formed German sentence.
There are some minor exceptions to this rule - "regnen", "schneien" and other verbs relating to weather activities and similar "realm conditions" (e.g. "es stinkt") definitely need "es" as a subject, even if V2 placement could otherwise be met.
The fact that "es" is used only to ensure the V2 condition (rather than being the subject) can be checked in many cases by replacing it with the particle "da" or the adverb "hier" - If the sentence still works, the assumption is true - This BTW works in your example as
Da sollte daran erinnert werden, dass ...
On the other hand, there are cases where "es" is simply not allowed in sentences - this is mainly the case for sentences using an impersonate passive like in
Auf den Fluren darf [no
es!] nicht gesungen oder musiziert werden
This sentence definitely wouldn't work with any subject whatsoever - This shows that German sentences are perfectly fine with having no subject at all as long as the V2 condition is met.