In the play Romeo and Juliet by William Shakespeare the author shows us how quickly a relationship can run wild and precipitate events when the sweethearts are very young. It is because impulsiveness, youth and passion are a very flammable mixture. The Friar can see these issues and is worried that they are rushing into a more serious stage of the relationship before anyone is ready for it - including them. The problem is that although he sees pitfalls ahead, he sees the future through rose-tinted spectacles of idealism - he also gives himself airs in thinking he can change the world so easily in using the couple as a tool to end the feud. As a holy man he also has concerns about the illicit nature of the romance and probably wants to put it on a more pious footing by marrying them as a sacrament.
To me, Friar Lawrence's concerns about the relationship between these two is that they are totally rushing into it. I think you can sum up his concern in the following line. It comes from Act II, Scene 3. He says to Romeo:
Wisely and slow; they stumble that run fast.
Friar Lawrence is concerned because up until this point, Romeo had been so hopelessly in love with Rosaline. The friar could not believe that Romeo could really fall in love with someone else so quickly. He counseled patience. Too bad they didn't listen to him... maybe they would not have died if they had taken it a bit slower.
Friar Laurence may start out with some concern about this relationship but I don't think it last very long. It is his pride that becomes the central focus. He believes that if he joins the two in marriage that it will end the feud and he will achieve fame with this result.
There are many points in this drama where Friar Laurence could have said, "enough this has gone too far. Let's confess what we have done and accept the consequences of our actions." However, he continues with his plan because of his pride. It be argued that he is only concerned with himself.