What are some quotes from Friar Laurence that indicate his loyalty to Romeo?

In Act 2, Scene 3, when Romeo tells Friar Lawrence he hasn't been to bed yet, the friar says, "God pardon sin! Wast thou with Rosaline?" The fact that Lawrence knows Romeo so well is something that lays the foundation for loyalty. So does the fact that his response starts with forgiveness rather than judgment. Likewise, when he listens to Romeo talk and talk about how he now loves Juliet in that same scene, Lawrence chides him for changing quickly, but doesn't really judge him harshly. In Act 2, Scene 6, Lawrence speaks kindly to Juliet, saying, "Romeo shall thank thee, daughter, for us both." This shows loyalty by aligning Lawrence's fate with Romeo. You can really see loyalty starting in Act 3, Scene 3, when Lawrence keeps Romeo from killing himself. These lines start with "Hold thy desperate hand!"
In that same scene, Lawrence chides Romeo for how he changes who he loves, saying "Holy Saint Francis, what a change is here! Is Rosaline, whom thou didst love so dear, So soon forsaken? young men's love then lies
Not truly in their hearts, but in their eyes."
 
However, despite judging Romeo for this change, he soon thereafter offers to help him, saying, "But come, young waverer, come, go with me, In one respect I'll thy assistant be;"
If you disagree with someone's actions, but help them anyway, that's a sign of loyalty.
 
 

Friar Laurence has a pivotal role in Romeo and Juliet: he marries the couple in secret, he helps Romeo escape to Mantua, and he helps Juliet fake her own death to avoid marrying Paris. It is his plan that goes awry, ultimately resulting in the couples’ deaths; but despite the outcome, Friar Laurence is quite loyal to Romeo.
Friar Laurence hopes that by marrying Romeo and Juliet, he can put an end to the feud between the Montagues and the Capulets:

I’ll thy assistant be
For this alliance may so happy prove
To turn your households’ rancour to pure love (II.ii.90-93)

Still, his loyalty to Romeo extends beyond their union. In Act 2, Scene 2, we can see that he has been giving Romeo advice about Rosaline:

Romeo: Thou chidd’st me oft for loving Rosaline
Friar Laurence: For doting, not for loving, pupil mine (II.ii.181-82)

This shows that Romeo is comfortable confiding in Friar Laurence, and the affectionate term “pupil” suggests a close relationship. After Romeo is banished, the Friar hides him (against the Prince's orders) and encourages him not to despair: 

What, rouse thee, man! thy Juliet is alive,
For whose dear sake thou wast but lately dead (III.iii.134-35)

Friar Laurence comes up with a plan to save Romeo and reunite him with Juliet. Obviously, it goes wrong (the Friar is in many ways an earthly, bumbling character, despite his title), but his intentions are good:

But look thou stay not till the watch be set,For then thou canst not pass to Mantua;Where thou shalt live, till we can find a timeTo blaze your marriage, reconcile your friends,Beg pardon of the prince, and call thee back (III.iii.147-151)

At Friar Laurence’s suggestion, Romeo flees for Mantua, while the Friar promises to beg pardon from the prince on his behalf. While Romeo is away, he tries to help Juliet get out of her engagement to Paris. In Act 4, Scene 1 he tries to dissuade Paris from marrying Juliet:

You say you do not know the lady's mind?Uneven is the course. I like it not (IV.i.4-5)

When Paris proves himself determined, the Friar helps Juliet escape by faking her own death:

Take thou this vial, being then in bed,And this distilled liquor drink thou off;When presently through all thy veins shall runA cold… (IV.i.93-96)

He sends a letter to Romeo explaining the circumstances, but the letter carrier, Friar John, is delayed. Realizing the mistake, he goes to the Capulet family tomb to retrieve Juliet, but Romeo is already dead.
When his plan ultimately goes wrong, his guilt and sadness are evident:

If aught in thisMiscarried by my fault, let my old lifeBe sacrificed… (V.iii.265-67)

His (presumably) long relationship with Romeo, his instrumental role in Romeo and Juliet’s marriage, and his good intentions in spite of the outcome show Friar Laurence’s loyalty to Romeo throughout the play. 
Source: Shakespeare, William. “Romeo and Juliet.” The Norton Shakespeare: Tragedies. 2nd ed. Eds Stephen Greenblatt et al. New York, NY: W. W. Norton & Company, Inc., 2006. 189-256. Print.