A Tantalus (q.v.).
In the full flood stands Tantalus, his skin Washed o'er in vain, for ever dry within. He catches at the stream with greedy lips - From his parched mouth the wanton torrent slips ... Change but the name, this fable is thy story: Thou in a flood of useless wealth dost glory, Which thou canst only touch, but never taste.
Cowley: Horace, satire i.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894