Quine's argument about existence relies on Russell's theory of descriptions. Therefore, one source of criticisms on Quine's thesis are criticisms on the theory of descriptions.
The theory of descriptions (On Denoting (1905)) was motivated by a desire to avoid admitting non existing objects. Russell's famous example was "the king of France is bald". The expression "the king of France" seems to refer to an object. But since there is no king in France, what does the expression refer to? Now according to Russell's theory, the sentence "the king of France is bald" does not refer to any objects at all. Instead of reference, there is an underlying quantification ("there is a king in France, and..").
Quine (On What There Is (1948)) merely took the theory of descriptions one step further. He argued that Russell's move applies to every name and referring expression, not just to problematic ones like "the king of France". So Quine, following Russell, eliminated all references to objects in favor of quantifications over classes of objects, as the way in which language connects to the world. That is the meaning of the motto "to be is to be a value of a bound variable".
The theory of descriptions received various criticisms. The most famous work, in this regard, is probably Saul Kripke's Naming and Necessity (1980). Kripke explored the functions of names, and attacked Russell's and Quine's thesis that a name can be replaced by a description, without remainder. Several contexts reveal deep differences between names and descriptions. Some of these contexts are modal (related to necessity and possibility).
Kripke's criticism has been received as very effective. After it, The descriptive theory of names has been no longer widely accepted. Philosophers got back to the (apparently) more common-sensical view that names do refer to objects. This, however, reopened the problem of non-existing objects.