Can I add a responsible person as trustee to my gun trust who lives in another state?

Yes.  You can add a responsible person/co-trustee to your NFA gun trust who lives in another state.  The trustee that lives in the different state will not however be able to use and possess the NFA firearms in a state that does not allow possession of the NFA firearms. The trustee will only be able to use the and possess the NFA firearms in a state where you can legally posses the NFA firearms.  The NFA laws vary from state to state.  You will need to check about your local NFA firearm laws to see which apply. 

In order to transport the NFA firearms to another state you will also need to fill out the "Application to Transport Interstate or to Temporarily Export Certain National Firearms Act (NFA) Firearms", which is the ATF Form 5320.20.  You will need approval from the ATF before you transport the NFA firearms across state lines.