There must be a difference in pronunciation between "dörr" and "dör", right? Or is it just one of those things where context is what differentiates them in speech?
Yes, dör has a long vowel and a short consonant, whereas dörr has a short vowel and a long consonant. It can be heard mostly in the vowel duration.
"Öppna din dörr, och säg att du vill ha mig här..." Made me think of that Swedish song.