It seems both
(mapcar \'car \'((foo bar) (foo1 bar1)))
and
(mapcar #\'car \'((foo bar) (foo1 bar1)))
w
Try passing an anonymous function (lambda) to your mapcar and you'll see that #' is required since the quote by itself expects a symbol that is bound to a function, but the symbol doesn't exist in an un-named function:
CL-USER> (mapcar '(lambda (x) (format t "it is ~d" x)) (list 3 5 7))
; Evaluation aborted on #.
vs:
CL-USER> (mapcar #'(lambda (x) (format t "it is ~d" x)) (list 3 5 7))
it is 3it is 5it is 7
(NIL NIL NIL)