Yes, poly, or any oil-based resin varnish, will adhere to the cured tung oil. The oil has polymerized into a film itself (within and on the surface), just like the cured varnish would (although minus the resin). Do a light, fine grit sanding (180, 220, or as recommended on the poly can), then apply the poly as directed. Do note, though, that in a high use/hard use area as a kitchen countertop, the poly may need re-coating in time. This would just be another scuff sanding and re-apply a coat or two. Thin the poly up to 50/50 with mineral spirits for easy wipe-on use. Odors should decrease greatly after a couple days even if full hardness cure may take a few weeks.
Is your metal faucet in direct contact with the wood? I think cherry has tanic acids in it that could react with metals, but faucet metal is typically brass, not iron-based ferrous metals, which I think(?) are most reactive to tanic acid from woods like oak and others. Anyone else know for sure? (Brass screws are used in oak to limit black stains.) Of course just daily normal use water splashes probably don't help right there at the base of the faucet. Even if wiped up right away, a bit of moisture can hang out right at the faucet point of contact with the wood.
You may want to consider the sheen of the varnish to best match the lower sheen of the oiled surface. Gloss poly, which is usually toughest, will have the high gloss appearance unless buffed back.