Fission, 7 bytes

'!+!NR" 

A rather simple modification of the shortest Fission quine I have found so far: I'm simply using the non-destructive ! instead of O and added an N for newline.

So all in all, here is how it works: control flow starts at the R with a right-going atom. " toggles string mode, which means everything until the next " is printed: in this case '!+!NR. That leaves the " and the newline to be printed. '! sets the atom's mass to 33, + increments it to 34 (the character code of ") and ! prints the quote. N prints a newline, and R is now a no-op in this case, so the loop starts over.

The following 7-byte solution also works:

"NR'!+!