"The bed sheet is dirty, but I sleep with it."
Translation:Sengetøjet er beskidt, men jeg sover med det.
No, as that would be "but I sleep in it", which would be like saying that you are wearing the bedding. You can say that about a pyjama, for example.
"Pyjamassen er beskidt, men jeg sover i den" (the pyjama is dirty, but I sleep in it.)
The only correct phrase is "men jeg sover med det"