If the formula is open, it can also contain free variables. Technically, $x=y$ doesn't contain a quantifier. Whether or not you want to allow that case or not depends on why you're interested in quantifier-free formulas in the first place.
If you disallow free variables, then you can, in a way, interpret a quantifier-free formula as a propositional formula. You can simply treat its atomic parts, i.e. a predicate applied to some expression containing functions and constants, as propositions. If propositional logic proves such a translation to be unsatisfiable, then first-order logic will agree.
If propositional logic proves it to be satisfiable, then first-order logic only necessarily agrees if you look at the formula alone. But if you look at it in the context of some theory, then axioms of the theory can of course constraint the truth assignment to the atoms of the formula, and so the formula may be disproved by the theory, even if it's propositional translation is satisfiable.