What is the motivation/drive of friar lawrence?
There are two instances in which Friar Laurence acts upon his own judgments in Shakespeare's Romeo and Juliet. The first, of course, is in marrying Romeo and Juliet; the second is providing Juliet the vial to drink that will induce a death-like state. By giving Juliet the potion, the Friar hopes to (a) gain some time for the parents to calm down and prevent Juliet from killing herself in desperation by giving her an alternative choice:
If rather than marry Count Paris
You have the strength of will to kill yourself,
Then it is likely you will try
Something like death to chase this shame away
Something that will cover you with death himself to escape from it?
And, if you will take the dare, I'll give you the solution. (2.1.70-75)
The Friar also hopes to (b) force the parents to realize how much they love their daughter as they believe her dead. Friar Laurence hopes that when Juliet is "returned to life" Lord and Lady Capulet will be so overjoyed that she is alive that they will be reasonable about her marriage and seek to ameliorate their relationship with the Montagues. He promises Juliet that he will send for Romeo that she may return to Mantua with him when she awakens.
Now, to return to the first interference of Friar Laurence. It is stated by him that the Friar does, indeed, not want the couple to commit mortal sins by having sexual relations outside of marriage--
Come, come with me, and we will make short work,
For, by your leaves, you shall not stay alond
Till Holy Church incorporate two in one. (2.6.35-37)
However, it also stands to reason and is consistent with his thinking in the second situation that the Friar also hopes that if the chldren of the feuding families marry, then the parents may feel compelled to live amicably for the sake of their children's happiness as well as for that of possible grandchildren. He tells Romeo,
So smile the Heavens upon this holy act
That afterhours with sorrow chide us not! (2.6.1-2)
One of the contrasts between Friar Lawrence and most of the rest of the city is that he is supposed to be loyal to his parishioners, regardless of which family they are in, etc. He is also protected to a certain extent from the politics of the city because he is of the church, something held sacred by both families.
Because of this, I believe he acts mainly out of a genuine desire to see what is best happen for Romeo and Juliet and without a great deal of concern to the consequences of that union. He is of course also interested in their salvation so he is certainly willing to marry them before they commit a serious sin like fornication, which he must have known they were wont to do!
I think that there are a couple of main motivations behind Friar Lawrence's actions.
The first, and less important, of the two is doing what is right for Romeo. The Friar is clearly concerned with Romeo's morals and his behavior. We can see this in how he is concerned about Romeo being too hasty to fall in love, for example.
The second of the two motivations is a desire for peace in the city. Perhaps because he is a priest, Friar Lawrence wants the Capulets and Montagues to stop fighting one another. This is why he marries Romeo and Juliet and why he helps her avoid having to marry Paris.