This is a very good question. In The Waste Land, Eliot quotes from Shakespeare several times, with most of the quotations coming from or allusions to The Tempest.
For instance, lines 48 and 125 of The Waste Land
(Those are pearls that were his eyes. Look!)
Are taken verbatim from Act 1, scene 2 of The Tempest:
Full fathom five thy father lies:
Of his bones are coral made:
Those are pearls that were his eyes:
An allusion to The Tempest is in The Fire Sermon section:
On a winter evening round behind the gashouse
Musing upon the king my brother's wreck
And on the king my father's death before him.
White bodies naked on the low damp ground
And bones cast in a little low dry garret,
Rattled by the rat's foot only, year to year.
This passage is intended to make the reader think of Ferdinand's speech, again in Act 1, scene 2:
Some god o' th' island. Sitting on a bank,
Weeping again the king my father's wrack,
This music crept by me upon the waters,
Allaying both their fury and my passion,
With its sweet air: thence I have follow'd it,--
Or it hath drawn me rather,--but 'tis gone.
In the section A Game of Chess, the reference to Cleopatra's "chair she sat……throne" is based on Enobarbus’s description of Cleopatra in Shakespeare’s Antony and Cleopatra. Lines 280-85 also allude to Antony and Cleopatra.
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In addition to the references to The Tempest and Anthony and Cleopatra, there are other parts of Eliot's poem that call on Shakespeare.
You can see echoes of Titus Andronicus in the poem's opening lines. (See this link for more details.)
In the flower giving and wet hair of the hyacinth girl, I see echoes of Ophelia, with her flowers and her death.
The "handful of dust" line may be an echo of Hamlet's riff on man being little more than dust.
And, of course, there's direct reference here:
O O O O that Shakespeherian Rag—
It's so elegant
'What shall I do now? What shall I do?'
This refers to Hamlet's indecision.