Common Lisp 667 657 645 Chars
My first attempt at code golf, so there are probably many tricks that I don't know yet. I left some newlines there to keep some residual "readability" (I counted newlines as 2 bytes, so removing 6 unnecessary newlines gains 12 more characters).
In input, first put the shapes then the field.
(let(b(s 0)m(e'(0 1 2 3 4 5 6 7 8 9)))
(labels((o(p i)(mapcar(lambda(j)(+ i j))p))(w(p r)(o p(* 13 r)))(f(i)(find i b))
(a(&aux(i(position(read-char)"IOZTLSJ")))(when i(push(o(nth i'((0 13 26 39)(0 1 13 14)(0 1 14 15)(0 1 2 14)(0 13 26 27)(1 2 13 14)(1 14 26 27)))(read))m)(a))))
(a)(dotimes(i 90)(if(find(read-char)"#=")(push i b)))(dolist(p(reverse m))
(setf b`(,@b,@(w p(1-(position-if(lambda(i)(some #'f(w p i)))e)))))
(dotimes(i 6)(when(every #'f(w e i))(setf s(1+ s)b(mapcar(lambda(k)(+(if(>(* 13 i)k)13(if(<=(* 13(1+ i))k)0 78))k))b)))))
(dotimes(i 6)(format t"[~{~:[ ~;#~]~}]
"(mapcar #'f(w e i))))(format t"[==========]
~a0"s)))
Testing
T2 Z6 I0 T7
[ ]
[ ]
[ ]
[ ]
[ # # #]
[ ## ######]
[==========]
[ ]
[ ]
[ ]
[# ###]
[# ### ]
[##### ####]
[==========]
10
NIL
