Salt is neither an acid or a base, so it does not require any neutralization.

Apparently what has happened is that salty water got under the tile and moisture continues to carry the salt out. Any moisture under there will of course evaporate where it hits the air, i.e. at the grout joints, leaving salt behind.

I am not sure this process can be sped up as it depends on dissolving the underlying salt and then getting water to carry it up and evaporate. Here's a stab at it: You could try removing it with a MINIMAL amount of water - scrub with a damp sponge. Then add a very small amount of clean water just to wet the joints. Wipe off excess water. Run a dehumidifier in the area to draw moisture out. When salt collects, repeat.

EDIT: I just realized this may be a wall not a floor? Not uncommon to have efflorescence of salts on basement walls. Happens on some tile I mounted on concrete behind my woodstove. You will have trouble getting water to go IN to the joints if it's a vertical wall though.

How good is the foundation drainage outside the basement wall in this area? Reducing the infiltrating moisture should help.

And if you paint over a wall that's efflorescing, I doubt the paint is going to hold, regardless of whether it's salts from soil, concrete or water softener. It's the moisture that's the problem.

This is efflorescence. The moisture in the soil is coming through the concrete and carrying salts in the concrete to the surface.

There are treatments for that. You'll want to research those.