Ah, indeed, I tend to confuse it for Brainfuck which at least can do something of use.

I/O on a Turing machine consists of initializing the tape with your input and reading the output off the tape once it's finished computing.

Indeed, and that exactly puts all the onus of exploitability on the machine's environment. The tape might as well be written by regular expressions, if some of the outputs make the ‘interpreter’ do network requests and stuff.