Aluminum sheet as an underlay for electonic equipment
I don't think a plate is going to help very much for any of the things you are trying to do. If you actually need to prevent spread of flames, you would probably want to put metal under, and up the walls all around the NAS. But that would adversely affect wifi. If you do it anyway, I would not bother grounding the aluminum.

I don't see how putting aluminum plate or sheet under and appliance will help keep it cool. Especially in an adequately ventilated closet.

There are other materials besides metal for fire resistance such as gypsum board (which provides some limited resistance) and some rockwool materials with excellent fire resistance and insulation properties.

If you really need fire protection, I recommend using fire rated insulation. Roxul is one brand. If you use aluminum anyway, I suggest you put the aluminum between your NAS and the insulation. And I suggest you make sure there is a continuous insulation barrier between your NAS and any combustible material such as wood.

Note: I am not saying fire protection is required around an NAS. That is your premise. I am assuming it is true for the purposes of this answer. In reality, I am not sure you really need to put fire protection around an NAS. But if you do, you might as well do it right.