Our landlord is really flexible (says he doesn't mind what breed or size of dog, loves them all!) so we're lucky in that regard. We'll probably be buying a house after we leave this rental. The Dogs Home is happy to adopt to us, but I do understand why a lot of rescues would be cautious. The number of dogs on Gumtree, who are being given away by people who are moving, is terrible.
If we did have to move into another rental, I would never give up my dog. We'd find one that allowed it.