First is fine. It could also work with 'what was asked'. This will depend on when all this is happening. 'Don't do anything [today] that was asked of you [yesterday].' or 'Don't do anything [tomorrow] that is asked of you [today]." Second is awkward at best, wrong at worst. The word 'for' connects to the word 'do'. So technically, what you're saying is 'Don't do anything other than what you were asked for do'. That's obviously wrong. The correct way to say it would therefore be 'Don't do anything other than what you're asked to'.

The first one is correct. The second one still makes sense but is not as natural