What are some examples of Romeo being emotionally immature in Romeo and Juliet?

Though Romeo's age is never given in Shakespeare's play he is probably around 18 or 19. In today's world 18 year-olds do not marry 13 year-olds, but during the 16th century, it was perfectly acceptable. We also know that he is old enough to be able to kill a man, as witnessed by his slaying of Tybalt in Act III. 

From the beginning, Romeo displays behavior that could be considered immature. Benvolio tells Romeo's father...

Though Romeo's age is never given in Shakespeare's play he is probably around 18 or 19. In today's world 18 year-olds do not marry 13 year-olds, but during the 16th century, it was perfectly acceptable. We also know that he is old enough to be able to kill a man, as witnessed by his slaying of Tybalt in Act III. 

From the beginning, Romeo displays behavior that could be considered immature. Benvolio tells Romeo's father that he often sees Romeo alone in a grove of sycamores. Montague says in Act I, Scene 1,

Many a morning hath he there been seen,
With tears augmenting the fresh morning’s dew,
Adding to clouds more clouds with his deep sighs.
But all so soon as the all-cheering sun
Should in the farthest east begin to draw
The shady curtains from Aurora’s bed,
Away from light steals home my heavy son
And private in his chamber pens himself,
Shuts up his windows, locks fair daylight out,
And makes himself an artificial night.
Black and portentous must this humor prove,
Unless good counsel may the cause remove.

We find out that Romeo's depression is caused by his unrequited love for Rosaline, who spurns his advances and says she will live "chaste." Rather than accept this rejection and move on, Romeo dwells on the topic. Through a litany of oxymorons, Romeo complains to Benvolio Act I, Scene 1,

Why then, O brawling love, O loving hate,
O anything of nothing first create!
O heavy lightness, serious vanity,
Misshapen chaos of well-seeming forms,
Feather of lead, bright smoke, cold fire, sick health,
Still-waking sleep that is not what it is!
This love feel I, that feel no love in this.
Dost thou not laugh?

Of course, Romeo totally forgets Rosaline once he sees Juliet at Capulet's party. Later Friar Lawrence criticizes Romeo for falling in love with Juliet so quickly. He says in Act II, Scene 3,

Young men’s love then lies
Not truly in their hearts, but in their eyes.

Shakespeare wants us to believe Romeo is truly in love but, on the surface, we may see an immature young man who falls for the first pretty face he comes across. During the balcony scene, instead of taking a little time to get to know Juliet and consider the situation, Romeo leaps to a marriage proposal. For her part, Juliet is perfectly willing to wait, but, when she tries to adjourn to her bedroom he presses her for an admission of love and proposes.

They are married the very next day, even though Friar Lawrence warns Romeo to take things slow. The Friar says in Act II, Scene 6,

These violent delights have violent ends
And in their triumph die, like fire and powder,
Which, as they kiss, consume. The sweetest honey
Is loathsome in his own deliciousness
And in the taste confounds the appetite.
Therefore love moderately. Long love doth so.
Too swift arrives as tardy as too slow.

Romeo ignores the Friar's advice for the rest of his life. He is impetutous in everything he does. He ignores common sense, as well as the edict of the Prince, when he fights Tybalt to get revenge for the killing of Mercutio. Afterward he acts like a child as he whines to the Friar about being banished and unable to see Juliet. Romeo says in Act III, Scene 3,

’Tis torture and not mercy. Heaven is here
Where Juliet lives, and every cat and dog
And little mouse, every unworthy thing,
Live here in heaven and may look on her,
But Romeo may not.

In the final Act Romeo again exhibits his immaturity by plotting his own suicide after hearing of Juliet's death from Balthasar. Instead of confirming the news or checking with the Friar, Romeo launches himself into despair, procures poison, and rushes back to Verona to "lie with Juliet."

At almost every turn Romeo proves to be ill suited to handle the responsibility of love and marriage. He is impatient, prone to depression and willing to act on his slightest urge. Even though he is older he never acts with same level of courage and maturity which Juliet displays throughout the play.