Why didn't the eagles just take the dwarves and hobbit to Lonely Mountain?

Short Answer: They are not a taxi service plus its several hundred miles from the misty mountains to The Lonely Mountain.

Taxi!

In the book The Eagles are Gandalf's old friends and hearing the commotion see their buddy in trouble and come over to help.


Book excerpt: "What's all this uproar in the forest tonight?" said the Lord of the Eagles. He was sitting, black in the moonlight, on the top of a lonely pinnacle of rock at the eastern edge of the mountains. "I hear wolves' voices! Are the goblins at mischief in the woods?"

He swept up into the air, and immediately two of his guards from the rocks at either hand leaped up to follow him. They circled up in the sky and looked down upon the ring of the Wargs, a tiny spot far far below. But eagles have keen eyes 



The Eagles are also clever beings who help out Gandalf if he's in a dire situation. Not pack mules waiting around to cart dwarves.

Book excerpt: Eagles are not kindly birds. Some are cowardly and cruel. But the ancient race of the northern mountains were the greatest of all birds; they were proud and strong and noble-hearted. They did not love goblins, or fear them. When they took any notice of them at all (which was seldom, for they did not eat such creatures ), they swooped on them and drove them shrieking back to their caves, and stopped whatever wickedness they were doing. The goblins hated the eagles and feared them, but could not reach their lofty seats, or drive them from the mountains


The Eagles don't really care about the dwarves and their gold, additionally  they don't want to get shot with arrows.

 Book excerpt: The wizard and the eagle-lord appeared to know one another slightly, and even to be on friendly terms. As a matter of fact Gandalf, who had often been in the mountains, had once rendered a service to the eagles and healed their lord from an arrow-wound. So you see 'prisoners' had meant 'prisoners rescued from the goblins' only, and not captives of the eagles. As Bilbo listened to the talk of Gandalf he realized that at last they were going to escape really and truly from the dreadful mountains. He was discussing plans with the Great Eagle for carrying the dwarves and himself and Bilbo far away and setting them down well on their journey across the plains below.

The Lord of the Eagles would not take them anywhere near where men lived. "They would shoot at us with their great bows of yew," he said, "for they would think we were after their sheep. And at other times they would be right. No! we are glad to cheat the goblins of their sport, and glad to repay our thanks to you, but we will not risk ourselves for dwarves in the southward plains."

"Very well," said Gandalf. "Take us where and as far as you will! We are already deeply obliged to you. But in the meantime we are famished with hunger."


The Eagles dropped them off as far as they could and said good bye...

"Farewell!" they cried, "wherever you fare, till your eyries receive you at the journey's end!" That is the polite thing to say among eagles. "May the wind under your wings bear you where the sun sails and the moon walks," answered Gandalf, who knew the correct reply. And so they parted. And though the lord of the eagles became in after days the King of All Birds and wore a golden crown, and his fifteen chieftains golden collars (made of the gold that the dwarves gave them), Bilbo never saw them again-except high and far off in the battle of Five Armies. But as that comes in at the end of this tale we will say no more about it just now.