Nurse in Shakespeare's Romeo and Juliet can actually be considered more of a static character. Static characters are characters that actually don't change or develop throughout the story, as opposed to round characters that do develop and change, especially with respect to having self-realizations. However, whether or not she is a static character or one who evolves is actually open to some interpretation.
Nurse's one emotional drive as a character is pleasing Juliet. Nurse agrees to venture out into town to meet Romeo in order to acquire his plans for marriage simply because she loves Juliet and wants to do what she can to make her happy. We can actually even speculate that Nurse does not think very highly of Romeo. We see Nurse criticize Juliet's choice, even saying that she has made a poor choice in Romeo, in the lines,
Well, you have made a simple choice; you know
not how to choose a man. Romeo? No, not he. (II.v.39-40)
However, Nurse may be joking in those lines as she goes on to praise his looks saying,
Though his face be better than any man's, yet his leg excells all men's; and for a hand and a foot, and a body...they are past compare. (41-43)
In other words, Nurse is declaring that Romeo is better looking than any other man.
If Nurse truly is joking in saying that Romeo is a poor choice for Juliet, then one way in which we do see her change as a character is that she later changes her mind and praises Paris above Romeo, encouraging Juliet to marry Paris, as we see when she says,
I think it best you married with the County.
O, he's a lovely gentleman!
Romeo's a dishclout to him. An eagle, madam,
Hath not so green, so quick, so fair an eye
As Paris hath. (III.v.227-231)
I would actually argue that when Nurse criticizes Juliet's taste in Romeo she is speaking more honestly than she lets on in the next lines; Nurse truly does not think very highly of Romeo. I would also argue the Nurse's dislike of Romeo was of course increased when Romeo slayed Tybalt. Hence, Nurse is speaking honestly when she says that Paris is above Romeo. Hence, I think it can be argued that Nurse's opinion of Romeo stays static throughout the play and that she is a static character who wants nothing more than to please Juliet.
However, it is also possible to see her as having changed her mind about Romeo. If she has changed her mind, then we can see her as a character who has evolved from wanting to please Juliet to wanting to protect her.