I guess I really didn't define what "warm" mean....

Neutral papers render black as black. No tint of any kind.
Cold papers render black with slight hint of blue'ish tone.
Warm papers render black with slight hint of brown/red.

SOME warm tone paper has off-white, cream colored paper base so highlight takes warm tone. Ilford WTFB has quite a definite cream color.

Please note, it's just a HINT.... Sometimes it's to degree that you'll have hard time telling them apart unless you have prints side-by-side. If you want an obvious tone, you'll have to tone it.