Inspired by Gynvael Coldwind's 125 bytes version

Assemble with Yasm
non-compliant (dialect) version, non-commands are not ignored

[bits 16]
[org 0x100]
; assume bp=091e used
; assume di=fffe
; assume si=0100
; assume dx=cs (see here)
; assume cx=00ff
; assume bx=0000
; assume ax=0000 used (ah)
; assume sp=fffe
start:	mov	al, code_left - start
code_start:	mov	ch, 0x7f ; allow bigger programs
mov	bx, cx
mov	di, cx
rep	stosb
mov	bp, find_right + start - code_start ;cache loop head for smaller compiled programs
jmp	code_start_end
find_right:	pop	si
dec	si
dec	si ;point to loop head
cmp	[bx], cl
jne	loop_right_end
loop_right:	lodsb
cmp	al, 0xD5 ; the "bp" part of "call bp" (because 0xFF is not unique, watch for additional '[')
jne	loop_left
inc	cx
loop_left:	cmp	al, 0xC3 ; ret (watch for ']')
jne	loop_right
loop	loop_right ;all brackets matched when cx==0
db	0x3c ;cmp al, xx (mask push)
loop_right_end:	push	si
lodsw	; skip "call" or dummy "dec" instruction, depending on context
push	si
code_sqright:	ret
code_dec:	dec	byte [bx]
code_start_end:	db	'$' ;end DOS string, also "and al, xx"
code_inc:	inc	byte [bx]
db	'$'
code_right:	inc	bx ;al -> 2
db	'$'
code_left:	dec	bx
db	'$'
code_sqleft:	call	bp
db	'$'
; create lookup table
real_start:	sub	byte [bx+'>'], al ;point to code_right
add	byte [bx+'['], al ;point to code_sqleft
mov	byte [bx+']'], code_sqright - start
lea	sp, [bx+45+2] ;'+' + 4 (2b='+', 2c=',', 2d='-', 2e='.')
push	(code_dec - start) + (code_dot - start) * 256
push	(code_inc - start) + (code_comma - start) * 256
pre_write:	mov	ah, code_start >> 8
xchg	dx, ax
; write
mov	ah, 9
int	0x21
; read
code_comma:	mov	dl, 0xff
db	0x3d ; cmp ax, xxxx (mask mov)
code_dot:	mov	dl, [bx]
mov	ah, 6
int	0x21
mov	[bx], al
db	'$'
db	0xff ; parameter for '$', doubles as test for zero
; switch
xlatb
jne	pre_write
; next two lines can also be removed
; if the program ends with extra ']'
; and then we are at 96 bytes... :-)
the_end:	mov	dl, 0xC3
int	0x21
int	0x20
compliant version, non-commands are ignored, but 104 bytes long

[bits 16]
[org 0x100]
; assume bp=091e used
; assume di=fffe
; assume si=0100
; assume dx=cs (see here)
; assume cx=00ff
; assume bx=0000
; assume ax=0000 used (ah)
; assume sp=fffe
start:	mov	al, code_nothing - start
code_start:	mov	ch, 0x7f ; allow bigger programs
mov	bx, cx
mov	di, cx
rep	stosb
mov	bp, find_right + start - code_start ;cache loop head for smaller compiled programs
jmp	code_start_end
find_right:	pop	si
dec	si
dec	si ;point to loop head
cmp	[bx], cl
jne	loop_right_end
loop_right:	lodsb
cmp	al, 0xD5 ; the "bp" part of "call bp" (because 0xFF is not unique, watch for additional '[')
jne	loop_left
inc	cx
loop_left:	cmp	al, 0xC3 ; ret (watch for ']')
jne	loop_right
loop	loop_right ;all brackets matched when cx==0
db	0x3c ;cmp al, xx (mask push)
loop_right_end:	push	si
lodsw	; skip "call" or dummy "dec" instruction, depending on context
push	si
code_sqright:	ret
code_dec:	dec	byte [bx]
code_start_end:	db	'$' ;end DOS string, also "and al, xx"
code_inc:	inc	byte [bx]
db	'$'
code_right:	inc	bx ;al -> 2
code_nothing:	db	'$'
code_left:	dec	bx
db	'$'
code_sqleft:	call	bp
db	'$'
; create lookup table
real_start:	inc	byte [bx+'<'] ;point to code_left
dec	byte [bx+'>'] ;point to code_right
mov	byte [bx+'['], code_sqleft - start
mov	byte [bx+']'], code_sqright - start
lea	sp, [bx+45+2] ;'+' + 4 (2b='+', 2c=',', 2d='-', 2e='.')
push	(code_dec - start) + (code_dot - start) * 256
push	(code_inc - start) + (code_comma - start) * 256
pre_write:	mov	ah, code_start >> 8
xchg	dx, ax
; write
mov	ah, 9
int	0x21
; read
code_comma:	mov	dl, 0xff
db	0x3d ; cmp ax, xxxx (mask mov)
code_dot:	mov	dl, [bx]
mov	ah, 6
int	0x21
mov	[bx], al
db	'$'
db	0xff ; parameter for '$', doubles as test for zero
; switch
xlatb
jne	pre_write
; next two lines can also be removed
; if the program ends with extra ']'
; and then we are at 100 bytes... :-)
the_end:	mov	dl, 0xC3
int	0x21
int	0x20