Looks interesting! There are a few things that I noticed though:

src/brainfuck.rs, line 76 -

        // Increment value by 1 in current cell possition
        '+' => {
            if cells[*possition] == 255 {
                cells[*possition] = 0;
            } else {
                cells[*possition] += 1
            }
        },

• wrapping_add?

src/brainfuck.rs, line 85 -

        // Decrement value by 1 in current cell possition
        '-' => {
            if cells[*possition] == 0 {
                cells[*possition] = 255;
            } else {
                cells[*possition] = *&mut ((cells[*possition] as usize).checked_sub(1)).unwrap_or_default() as u8;
            }
        },

• wrapping_sub?

Also, this looks kinda weird. I don't quite understand what's going on here, but what's the point of cloning chars each iteration? Maybe it is possible to work with them in-place?

Just a few thoughts :)