What role did greed play in Tolkien's books?

[Saurondil]

A pretty prominent one. Examples include:

Melkor
The Dwarves as a People
Feanor
Thingol
Turgon
Ancalime
Tar-Atanamir the Great
Tar-Ciryatan
Tar-Telemmaite
Ar-Pharazon
the Numenoreans after the Shadow falls.
Sauron
Isildur ?
Smaug
Thorin has trouble with the Arkenstone, at least
Denethor II
Lobelia Sackville-Baggins

- and arguably Saruman.

Galadriel & Gandalf are both tested by having to resist their desire for the Ring.

All these characters desire, or are tempted to desire, what they cannot have, or should not use as they do. Melkor wants "the dominion of Arda". Feanor comes to love the Silmarils with a "possessive love", so that he cannot give them up after the Trees are poisoned. Denethor is so intent on "the good of Gondor", and his grief at the loss of Boromir, that he loses sight of the big picture. And Turgon is so enamoured of the beauty and strength of Gondolin, that he fails to heed the warning of Ulmo. Lobelia covets Bag End - with disastrous results

IMHO, the motif of greed could be regarded as a motif of disordered love - and love is a very prominent motif in the books. This is one reason Tom Bombadil is such a very important character - he has the inner freedom that protects him from wrongful desire. A Vala & a Hobbit and many beings between all suffer from wrongful desires of various desires - he is almost defined by not doing so. He is anything but a meaningless excrescence added for no good purpose - he is essential to the moral structure of the story.

I think making is essentially a form of self-giving, of - in a sense - relinquishing control, stepping back so that what is made can have a life of its own. And I think this requires power no Valar can have, because they are limited, not transcendent. FWIW, AFAICS making = subcreation, with creation in the strict sense being something only Eru can do. For creation, one must have the Flame Imperishable.