Should I put down underlayment when putting a layer of tile on top of another?
Per the Armstrong subfloors and underlayment manual, all Armstrong vinyl tile installations require an underlayment.

If you don't have an approved underlayment, you may not receive full warranty coverage for the tiles.

Further, the underlayment is secured using mechanical fasteners through the old tiles to the subfloor, so you don't have to worry about whether the adhesive will secure the new tiles to the old tiles. The adhesive will bond correctly with an appropriately chosen underlayment, and the tiles will not bubble or spread due to temperature fluctuations and differing rates of expansion between the new tile and the old.  This can only be guaranteed when the adhesive works, and most tile surfaces are designed to be difficult to stick to or stain, so finding a suitable adhesive might be difficult.