Interpret brainf***

Python (no eval), 317 bytes

from sys import*
def f(u,c,k):
 while(c[1]>=k)*u:
  j,u='[]<>+-,.'.find(u[0]),u[1:];b=(j>=0)*(1-j%2*2);c[1]+=b*(j<2)
  while b*c[c[0]]and j<1:f(u,c,k+1);c[1]+=1
  b*=c[1]==k;c[[0,c[0],2][j/2-1]]+=b
  if(j==6)*b:c[c[0]]=ord(stdin.read(1))
  if(j>6)*b:stdout.write(chr(c[c[0]]))
f(open(argv[1]).read(),[-1]+[0]*30003,0)

brainfuck, 843 691 bytes

Edit: decided to revisit this and found a surprising number of ways to golf off bytes

>>>,[>++++[-<-------->]<-[>+<<]>[----------[>]>[<+<+>>>>]<<<-[>]>[<+<+>>>>]<<<-[>]>[<+<+>>>>]<<<-[>]>[<-<+++>>>>]<<<--------------[>]>[<++<+>>>>]<<<--[>]>[<-<+++++++>>+>>]<<++++[-<------>]+<+[>]>[<++<+>>>>]<<<--[>]>[<<+>>>>]<<-<[+]<[>]>,>]<]<-[<]>[-[<<]>[<+[>]>>[<+[<<[<]<<-[>>]<[>>>>[>]>+<<[<]<]<-[>>]<[>>>>[>]>-<<[<]<]<++[->>+<<]>>[>]>]]<<<[<]>-<]>-[<<]>[<++[>]>+>[<-]<[<<[<]>[-<<+>>]>--[<<]>[[>]>+<<[<]<]>+[<<]>[[>]>-<<[<]<]>+[>]>]<<[<]>--<]>-[<<]>[[>]>>.<<<[<]<]>-[<<]>[[>]>>-<<<[<]<]>-[<<]>[[>]>>,<<<[<]<]>-[<<]>[[>]>>+<<<[<]<]>-[<<]>[[>]>>>[>>]>[<<<[<<]<+>>>[>>]>-]>[-]<<+[<[->>+<<]<]<[->>+<<]<[<]<]>-[<<]>[[>]>-[+>[-<<+>>]>]+<<[-]+[-<<]<[->>>[>>]>+<<<[<<]<]<[<]<]<++++++++>>[+<<->>]>]

This takes input in the form code!input where the !input is optional. It also simulates negative cells without using negative cells itself and can store up to (30000-(length of code+6))/2 cells.

Try it online!

16 bit 8086 machine code: 168 bytes

Here's the base64 encoded version, convert and save as 'bf.com' and run from Windows command prompt: 'bf progname'

gMYQUoDGEFKzgI1XAgIfiEcBtD3NIR8HcmOL2LQ/i88z0s0hcleL2DPA86sz/zP2/sU783NHrL0I
AGgyAU14DTqGmAF194qOoAH/4UfDJv4Fwyb+DcO0AiaKFc0hw7QBzSGqT8MmODV1+jPtO/NzDaw8
W3UBRTxddfJNee/DJjg1dPoz7U509YpE/zxddQFFPFt18U157sM+PCstLixbXUxjTlJWXmV+

EDIT

Here's some assembler (A86 style) to create the executable (I had to reverse engineer this as I'd misplaced the original source!)

    add dh,10h                              
    push dx                                 
    add dh,10h                              
    push dx                                 
    mov bl,80h                              
    lea dx,[bx+2]                         
    add bl,[bx]                            
    mov [bx+1],al                         
    mov ah,3dh                              
    int 21h                                 
    pop ds                                 
    pop es                                 
    jb ret                               
    mov bx,ax                              
    mov ah,3fh                              
    mov cx,di                              
    xor dx,dx                              
    int 21h                                 
    jb ret                               
    mov bx,ax                              
    xor ax,ax                              
    repz stosw                                     
    xor di,di                              
    xor si,si                              
    inc ch                                 
program_loop:
    cmp si,bx                              
    jnb ret                               
    lodsb                                    
    mov bp,8                            
    push program_loop
symbol_search:                       
    dec bp                                 
    js ret
    cmp al,[bp+symbols]
    jnz symbol_search
    mov cl,[bp+instructions]
    jmp cx                                 
forward:
    inc di                                 
    ret                                    
increment:
    inc b es:[di]                      
    ret                                    
decrement:
    dec b es:[di]                      
    ret                                    
output:
    mov ah,2                              
    mov dl,es:[di]                            
    int 21h                                 
    ret                                    
input:
    mov ah,1                              
    int 21h                                 
    stosb                                    
backward:
    dec di                                 
    ret                                    
jumpforwardifzero:
    cmp es:[di],dh                            
    jnz ret                               
    xor bp,bp
l1: cmp si,bx                              
    jnb ret
    lodsb                                    
    cmp al,'['                              
    jnz l2
    inc bp
l2: cmp al,']'                              
    jnz l1
    dec bp                                 
    jns l1
    ret                                    
jumpbackwardifnotzero:
    cmp es:[di],dh                            
    jz  ret
    xor bp,bp
l3: dec si                                 
    jz  ret
    mov al,[si-1]                         
    cmp al,']'
    jnz l4
    inc bp  
l4: cmp al,'['                              
    jnz l3
    dec bp                                 
    jns l3
    ret                                    
symbols:
    db '><+-.,[]'
instructions:
    db forward and 255
    db backward and 255
    db increment and 255
    db decrement and 255
    db output and 255
    db input and 255
    db jumpforwardifzero and 255
    db jumpbackwardifnotzero and 255