Moving my answer as well.

I’d say “toucher à” is used when you’re not supposed to touch.

If you can break something, change settings on a software, etc. It even work for flirting with someone you’re not supposed to : “Personne (ne) touche à ma copine, c’est compris ?“.

Ne me touchez pas. Don’t touch me. Transitive.
Ne touche pas la casserole. Elle est chaude. Transitive.

Ne touchez pas à ces papiers. Don’t mess with these papers {touch, disturb, mess with, involve yourself with}

When it’s the idea of mess with or involve yourself with: toucher à. Intransitive.

http://www.cnrtl.fr/definition/toucher