I would use your two phrases as follows in example conversations. My friend says: "Last week, Joe's friends called at his house to ask him to help them, but he was away at work" I say: "Did they forget that Monday is a working day?" My friend says: "Joe's friends have just gone to his house to ask him to help." I say: "Have they forgotten that Monday is a working day?"

They are both correct but they're used to talk about different situations. Did they forget ... refers to a past time. It is asking whether they had forgotten at some point in the past but it doesn't tell us about the present. Have they forgotten ... refers to the present time. It is asking if they currently don't remember that Monday is a working day. Examples could be: [On Tuesday] Why didn't they call us yesterday? Did they forget that Monday is a working day? [On Monday] Why haven't they called us yet? Have they forgotten that Monday is a working day?

Hello there! Honestly both sound 100% natural to me! But the first option is likely the one more used!

Both are correct. I would say that “Have they forgotten…”’is slightly more formal. “Did they forget…” is a sentence I might expect to hear in spoken English. I would probably also say “a work day” rather than “a working day.”