The present perfect, 'have forgotten', tells you about the situation at the present time. We use this form to talk about something that has an impact on the present moment. If you say 'I've forgotten it', this is describing a problem that you have now. For example, if you have forgotten some information, this means that you are unable to recall this information at the moment. If you have forgotten to bring an object, it means that you don't have this thing with you now.
By contrast, 'I forgot it' only tells us something about the past. We don't know whether you later remembered this information or not, or whether you went back to get the thing that you'd forgotten. We know that you forgot something - but we've no idea what happened next.