The relationship between Bilbo and Thorin changes when Thorin realizes that Bilbo has been hiding the Arkenstone from him.

Bilbo is having a terrible time.  None of the other dwarves dare to contradict Thorin, so the siege of the mountain continues.  This does not please Bilbo at all.

Bilbo, of course, disapproved of the whole turn of affairs. He had by now had more than enough of the Mountain, and being besieged inside it was not at all to his taste. (Ch. 15)

Biblo is also worried that Thorin is looking for the Arkenstone.  He has the dwarves search around for it, and Bilbo does not want to think of what would happen if they discover he has it hidden under his pillow.  The Arkenstone is very valuable, and it also holds sentimental value to Thorin.  Biblo will be in big trouble if Thorin realizes he has it.

Biblo believes the Arkenstone is more valuable as a bargaining chip. He offers it up to the Elvenking, who is definitely surprised that he has it.

"But how is it yours to give?" he asked at last with an effort.

"O well!" said the hobbit uncomfortably. "It isn't exactly; but, well, I am willing to let it stand against all my claim, don't you know. … (Ch. 16)

Gandalf is impressed, but Bilbo is still worried.  Thorin is “stricken dumb with amazement and confusion” when Bard offers up the stone, and accuses the men of thievery.  In face of Thorin’s rage, Bilbo, despite his “dreadful fright,” gathers up the courage to admit he is the one that gave it to them.

Bilbo knows that Thorin will see this as the ultimate treachery.  Thorin was just beginning to trust and appreciate him, and now this happens.  Unfortunately, Bilbo does not think Thorin will act rationally.  He is used to being in charge, and getting his way.  No one really stands up to him except for Gandalf.

When Gandalf comes to Bilbo’s defense, Thorin accuses them both of betraying him.  He tells Gandalf to come and get Bilbo or he will throw him down.  The other dwarves are sad to see Bilbo go, but Thorin is just angry.  Bilbo knows that their relationship has taken a serious blow, but he did what he had to do to avoid a war.

What changes Bilbo's relationship with Thorin during the mountain siege in The Hobbit?

Unfortunately, dwarves are not known for their gentleness of temper or forgiving natures.  Bilbo invokes Thorin's wrath when the dwarf realizes that his prize Burglar has, in fact, burgled his prize heirloom, the Arkenstone, right out of the hoard in the Lonely Mountain. 

Now, Bilbo did not undertake this action lightly, realizing the value of the Arkenstone and how precious it was to Thorin.  The hobbit gives the gem to Bard to aid in the stand-off between the dwarves, men, and elves over the treasure in the Lonely Mountain.  Thorin believed all of the treasure should belong to the dwarves, since they undertook the risk of retrieving it; the lake-men and elves feel that they too deserve a share as repayment for their losses during the dragon's terrible reign.  Bilbo tells Bard that the Arkenstone "is also the heart of Thorin.  He values it above a river of gold.  It will aid you in your bargaining" (244). 

When Bard presents the Arkenstone as a bargaining piece to Thorin, the dwarf becomes incredibly angry, accusing the lake-men of being thieves.  Bilbo, at this point, squeaks out that he gave it to them.  Thorin sees this as an utter betrayal, vowing "Never again will I have dealings with any wizard or his friends" (247). 

At the end of the battle as Thorin "lay wounded with many wounds," the proud dwarf calls for Bilbo and confesses "I wish to part in friendship from you, and I would take back my words and deeds at the Gate" (258).   Bilbo later reflects, "I wish Thorin were living, but I am glad that we parted in kindness" (259).