Does MPI_finalize release memory?

If you imagine the code that you've written without any of the MPI stuff, you'll see that each processor starts with a B array of size 1000, because you declare it as such:

integer b(1000)

Neither MPI_Init nor MPI_Finalise are involved in allocating or deallocating any of this memory.

Likewise, you can allocate an array at run time (C) and it will stick around until you explicitly deallocate it:

PROGRAM main
use mpi
implicit none

integer myr, numpr, ier
integer b(1000)
INTEGER, ALLOCATABLE :: C(:)



call MPI_INIT(ier)
call MPI_COMM_RANK(MPI_COMM_WORLD, myr, ier)
call MPI_COMM_SIZE(MPI_COMM_WORLD, numpr, ier)

ALLOCATE(C(1000))

if (myr .eq. 0) then
  b = 100  ! Set all values to 100
  c = 99   ! Ditto 99
endif 

call MPI_BCAST(b, 1000, MPI_INTEGER, 0, MPI_COMM_WORLD, ier)
call MPI_BCAST(c, 1000, MPI_INTEGER, 0, MPI_COMM_WORLD, ier)

call MPI_FINALIZE(ier)

PRINT *, myr, B(200)
PRINT *, myr, C(200)


DEALLOCATE(C)

END PROGRAM main

produces output:

           1         100
           1          99
           0         100
           0          99

Also, note that you had a typo (I think) in your initial code. You only send the first 100 members of B (which has size 1000).