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In Goethe's Faust, the protagonist can be considered a Romantic hero. (There are several characteristics of a Romantic hero. Use the enotes.com link below to see them all.)

Faust can be seen as a Romantic hero in Part I first because he is a character who does not conform to the norms of the time. Instead of actively pursuing an honest relationship with God, he makes a deal with Mephistopheles who serves the Devil. Faust also may be said to be a man of introspection. Strong emotion is also a characteristic of the Romantic hero.

But in Part II, Goethe has rejected Romanticism, and Faust is a Classical hero, though Faust continues, within the Classical structure, to show human emotion.

After Faust has built a harbor before "the wide shore's level shelf" and added a pleasure garden and canal on the newly exposed land, the peasant's "old hut" and "crumbling chapel" behind him make him "shudder" and ruin his "great estate." He tries to buy their land and give them a new home, but they refuse.  Faust then agrees with Mephistopheles,

Why bother yourself so much about them?
Shouldn’t you long ago have colonised them?

and says, "Then go and push them aside for me!" insisting they be relocated to "land ... / Set aside for the old folks."

But Mephistopheles and the Three Warriors alarm the peasants who "died of terror, peacefully," and burned on a "pyre," along with a "stranger," when the hut catches fire. The event makes Faust angry

Were you deaf to what I said?
I wanted them moved, not dead. ...
This mindless, and savage blow,
Earns my curse: share it, and go!

and remorseful ("Quickly said, too quickly done, I fear! –,"), especially when he seems to see the "Stars hide their faces, and their glow." 

Faust is then visited by Want, Debt, Distress, and Care. Care tells Faust that humans cannot find fulfilment ("Finding his fulfilment, never."). Faust is unconquered by Care:

Unholy spectre! So you hand our race
To the ravages of a thousand devils: ...
And yet, Care, I’ll not recognise you, nor even,
That creeping power of yours, by any token.

Care then blinds him (" Lifelong, all you men are blind, / Now, Faust, be so to the end!"). But Faust is paradoxically inspired:

The night seems deeper all around me,
Only within me is there gleaming light: 
I must finish what I’ve done, and hurry,

Hearing his grave being dug by Mephistopheles, Faust mistakenly believes work on his project is being carried out. Faust is newly inspired by encountering Care to complete the next phase of his project ("A swamp lies there below the hill,"). He is no long self-centered but has found a way to reproach Care through a new concern for the welfare of others: "Let me make room for many a million." The "gleaming light" within him--which ends the wager ("to the Moment I’d dare say: / ‘Stay a while! You are so lovely!’")--saves his soul from Mephistopheles, and he is taken into Heaven as he dies.

Characteristics of a Romantic hero:

http://www.enotes.com/topic/Romantic_hero

Why is Faust considered a Romantic Hero in Goethe's Faust?

Romanticism was a reaction against an emotionless rationality and structured literature that defined the Age of Enlightenment and Reason. Faust personifies this dichotomy on his way to becoming the Romantic hero. Faust starts out as a respected and admired representative of rationality and structure as a quiet-living academic who is an expert in every field of study from Mathematics to Divinity. Faust has spent his life in the pursuit of knowledge, which precluded (i.e., made impossible) involvement in life's pursuits of happineess, pleasure, and feelings.

Now, having found that the knowledge available to him is in the end analysis meaningless ("And see, that nothing can be known! / That knowledge cuts me to the bone."), he is searching for the knowledge of the cosmos "That [he] may detect the inmost force / Which binds the world, and guides its course." He wants to transcend the limits of earth's hold on mortal flesh and with the Moon

on mountains grand,
Amid thy blessed light ... stand, ... [and]
Float [as molecules] in thy twilight the meadows over,

After his encounter with Mephistopheles (Mephisto), Faust is persuaded to wager that Mephisto cannot lure him from his quest by vain pleasure and passion. His wager with Mephisto, which grows bit by bit from the initial,

Canst thou with lying flattery rule me,
Until, self-pleased, myself I see,—
Canst thou with rich enjoyment fool me,
Let that day be the last for me!

ends with a complicated challenge to Mephisto to tempt Faust and distract him through life's feelings:

Let us the sensual deeps explore,
To quench the fervors of glowing passion! ....
Then may delight and distress,
And worry and success,
Alternately follow, as best they can: ....

This is where Faust symbolically leaves behind the rational approach of the Age of Enlightenment and Reason and becomes the Romantic hero. In his new persona--as the Romantic, feeling, passionate hero--Faust engages in lust and murder and horror and despair when he seduces Gretchen (thanks to the motivation given by Mephisto's magical potion), then murders Gretchen's brother Valentine (while under Mephisto's control), then suddenly learns with rage about Gretchen's murders and imprisonment, then fails in earnest and heart-wrenching attempts to rescue her from execution. In Faust Part II, Faust becomes a Classical hero and is no longer a Romantic hero because Goethe renounced Romanticism in 1777 and pursued Classicalism thereafter.