Pronoun “you” can be omitted as subject in imperative form, what other pronouns can be omitted, when and why?

The pronoun you can be omited as a general rule, but sometimes I’ve seen sentences that should have used I or it as the subject but it was omitted.

In Standard English, the subject can be omitted only in an imperative sentence:

• Go to the store.

Colloquially, you can sometimes sneak by with an implied subject if there’s enough context:

• Got something to say?
• Went home sick — fell asleep immediately.
• Am not!

Leaving off the first word, or words, in a sentence is a common conversational strategy in English, explained here. It's called "Conversational Deletion" in the trade.

The deleted material often is, but need not be, a subject pronoun; any word or phrase that's predictable can be deleted, followed by the next predictable chunk, etc, until new information is reached. This is not random, but is governed by the usual syntactic rules, as explained in the dissertation mentioned in the link above.