There are several sources that may have provided inspiration for the "statue scene", as it is usually called, but no exact precursors.
Shakespeare's most obvious source is the story of Pygmalion in Ovid's Metamorphoses, which Shakespeare probably read in Arthur Golding's translation. The story can be found in Book X:
Now in the whyle by wondrous Art an image he did grave
Of such proportion, shape, and grace as nature never gave
Nor can to any woman give.
Art competing with nature is a theme that also occurs in The Winter's Tale: the supposed statue by Julio Romano is described in Act V, scene 2:
(...) a piece many
years in doing and now newly performed by that rare
Italian master, Julio Romano, who, had he himself
eternity and could put breath into his work, would
beguile Nature of her custom, so perfectly he is her
ape: (...)
Venus grants Pygmalion's wish for a wife as beautiful as his statue by bringing that statue to life. Golding's translation contains the following words:
In her body streyght a warmenesse seemd to spred.
This appears to be mirrored in Leontes's words about the "statue" (Act V, scene 3):
O, she's warm!
However, Hermione had only pretended to be a statue, whereas Galatea (the name used in sources other than the Metamorphoses) was a statue that had come to life. For this reason, it makes sense to look for additional sources of inspiration.
Another potential source is the story of Alcestis, which is best known through Euripides's play. The god Apollo had persuaded the Fates to allow king Admetus to live longer than the time allotted to him, on the condition that someone else would die in his stead. His wife Alcestis is the person who eventually agrees to die in his stead. After her death, Thanatos leads Alcestis away to the underworld, but Heracles intervenes and brings Alcestis back from the dead. Wikipedia's summary goes as follows:
When he [Heracles] returns, he brings with him a veiled woman whom he tells Admetus he has won in a competition. He asks his host to take her and look after her while Heracles is away on his labours. After much discussion, he finally forces a reluctant Admetus to take her by the hand, but when he lifts the veil, he finds that it appears to be Alcestis, back from the dead. Heracles has battled Death and forced him to give her up.
An important parallel between The Winter's Tale and Alcestis is that theatre goers are made to believe that the king's wife has died; from that point of view, both queens return from the dead, even though Hermione's dead was not real. (Pretending someone had died while keeping her hidden is a plot element that Shakespeare had already used in Much Ado About Nothing, i.e. in the Claudio-Hero plot.)
The removal of the veil in Alcestis may remind readers of Paulina drawing the curtain before Hermione's statue.
In Shakespeare: The Biography (London: Chatto & Windus, 2005), Peter Ackroyd mentions another potential source of inspiration (Chapter Seventy-Six). When James I entered London on 15 March 1604, Shakespeare and his fellow actors were probably part of the procession. Ackroyd writes (emphasis added):
There were seven triumphal gates, created in the style of Roman arches by Stephen Harrison; there were fountains, and flames, and living statuary. Shakespeare himself adopted the device of a statue coming alive, at a later date, in The Winter's Tale.
The plot element of a male character believing his wife dead for more than a decade (sixteen years in The Winter's Tale): Shakespeare had already used this in Pericles, Prince of Tyre, which he and George Wilkins wrote two or three years before The Winter's Tale. Pericles assumes his wife Thaisa dead for fourteen or fifteen years.
