"Help me close the door."
Translation:Hjælp mig med at lukke døren.
I am tasting them now... nam nam... I think the last one needs a comma, being imperative and all "Hjælp mig, luk døren!". But that does not convey the same meaning. You might keep it infinitive "Hjælp mig lukke døren". I think the original "Hjælp mig med at lukke døren" is best though. You may hear someone say what you suggest, but I do not think it is proper Danish. Some things are just denser in other languages. "Ayudame cerrar la puerta".
Thinking about it again, they might be accepted as answers... If you like :-) I am sorry for the non-answer.
Google results: "hjælp mig med": 1.660.000 "hjælp mig med at": 1.390.000 "hjælp mig at": 664.000 "hjælp mig lukke": 4 (but every verb should count here)