What problems of monastic life does the Friar speak of in "Fra Lippo Lippi" by Robert Browning?

The speaker in the poem is not a friar but a monk named Lippo Lippi—the "fra" in his name means "Brother," which is what a monk would be called.
Fra Lippo Lippi is caught outside the monastery walls by some guards. He has been drinking, which lowers his inhibitions, and he tells the guards about his life, including many complaints about the monastery. We learn it was poverty rather than vocation that called him to be a monk. He complains that he has to sneak around and climb out his window at night to have fun and that his superiors try to suppress his enjoyment of life's pleasures. He is a painter too and complains that his superiors don't like his realistic art, such as including the faces of real people they know in paintings to hang in church. They tell him:

Make them [church-goers] forget there's such a thing as flesh.
Your business is to paint the souls of men—

They want idealized images, just as they want Lippo Lippi to lead a pure, idealized life—and to Lippi, that isn't the way life or art are meant to be.