You would make a phosphate salt. Depending on its solubility, it may precipitate. Depending on the log K, your cola might no longer be acidic at all, and as others have suggested, not taste as nice, making the addition of both the phosphoric acid and calcium carbonate pointless.

2 H3PO4 + 3 CaCO3 -> Ca3(PO4)2 + 3 H2O + 3 CO2

The calcium phosphate phase listed above may not be the stable phase in cola solution at NTP, but is the simplest example.

How about intentionally adding a base that creates a very water-insoluble salt? Then the process is just 1) add base 2) filter 3) re-carbonate it, and you have non-acidic coke.

Yes, but it'd probably alter the flavor for the worse.

if you were interested in adding salt and water (the by products of a neutralization reaction) to the drink, then yes.

Salt? Sodium chloride? Or you mean the salt of phosphoric acid?

A salt, not salt.

Salt: sodium chloride (NaCl, table salt, etc.) A salt: an ionic compound from an acid-base neutralization