I am confused because of the addition of 세요 to the end. Since this means "please" I thought this could mean a polite request. However 그러지 마 sounds a strong,assertive way to tell someone to stop doing something. Would I be correct in thinking that the speaker is upset/annoyed despite the "세요"?
You can't say whether the speaker is annoyed or not without more information.
It is definitely not the most polite way to convey the idea though, because it means "Please don't do that", which is too direct and can sound like an order despite the polite ending form.
If you're saying it to a customer, for instance, you would use a softer sounding phrase, like telling them what might happen from whatever they're doing or just explaining the rule. For example, 진열장 물품은 만질 수 없게 돼 있어요 (It is a rule not to touch the displayed items) sounds more polite than 그러지 마세요.