What are some examples of the seven deadly sins in Lord of the Rings?
Seven Deadly Sins found in Lord of the Rings: Since your question specifically addressed lust, greed, wrath, addiction, temptation, I will focus on those five and provide examples and textual evidence:
Lust: One prominent example of lust in Lord of the Rings is Wormtongue's haunting watch over Eowyn in the Golden Hall. As Theoden is restored and Wormtongue removed from his position of influence, Gandalf remarks:
What was you promised price? When all the men were dead, you were to pick your share of the treasure, and take the woman you desire? Too long have you watched her under your eyelids and haunted her steps" (509).
Tolkien's characerization of Wormtongue reveal him to be a craven, envious man, longing for things beyond his grasp. His longing for Eowyn is a dark combination of lust and a desire to anger her brother, Eomer.
Greed: One example of greed (and also could fall under temptation) is when Pippin sneaks the Palantir to see it for himself:
"'I wish I had known all this before,' said Pippin. 'I had no notion of what I was doing.'
'Oh yes, you had,' said Gandalf. 'You knew you were behaving wrongly and foolishly; and you told yourself so, though you did not listen. [...] But if I had spoken sooner, it would not have lessened your desire, or made it easier to resist'" (584).
Another more powerful example of greed is Saruman's treachery in growing his forces of Uruk Hair and his destruction the trees in order to build his forces.
Wrath: Sauron, in many ways, is the picture of wrath. He seeks to overcome and control Middle Earth, but does so violently and with agression. His creatures under his domain, orcs and even the Nazgul, embody the darkness of rage and hatred through their cruel acts against all of the creatures of Middle Earth.
Temptation: The One Ring is the embodiment of temptation in Lord of the Rings; the evil piece of jewelry promises power, success, and fortune to all those who could possibly gain possession of it. Throughout the Fellowship of the Ring, Boromir struggles with temptation in regard to the Ring, which ultimately leads to his downfall at Amon Hen. Later, Frodo struggles with the increasing temptation of wearing the Ring as he progresses closer to Mordor.
Addiction: Gollum is the consummate picture of the evils of addiction. His obsession with the Ring both controls and ultimately destroys him. His long possession of it deep in the Misty Mountains has corrupted his mind, leaving him a wretched, vile creature, at war with himself over his desire for 'his precious.'