An emphatic phrase used with a negative sentence to mean 'by no means', 'under no circumstances', or 'never'.
間違っても人のメールを無断で開かないでください。
Under no circumstances open someone else's email without permission.
間違ってもその計画を一人で進めないで。上司に相談して。
Don't, under any circumstances, proceed with that plan alone. Consult your boss.
間違っても軽率な発言をしてはいけない場面だ。
This is a situation where you must not make careless remarks under any circumstances.