For some writers, the "untouchable" aspect creates an instant plot barrier: will he or won't he? I consider this plot problem a trifle dull. I also have a low opinion of the assumption that a "good" priest should give up his integrity and beliefs for love. In a remarkable Tony Hillerman novel, the priest ends up deliberately walking into a dangerous situation to escape pressure from the woman who is convinced he should give up all for love. What a jerk! And I mean the woman!
Rather than the priest who struggles with celibacy, I've always been more interested in the priest who struggles with his vocation or his theology. My priest, Rhys, from Anubis on Mars and Saint of Mars already has a lover, which is sort of/kind of okay but precludes him from rising further in the futuristic Catholic hierarchy. Since his lover is more sincerely orthodox than Rhys, Rhys's primary struggle is over the type of priest he will be.
CSI produced a decent-ish episode in which the priest-in-love came to terms with his affection by embracing other theological principles (God's call for him to be a husband and father) while Blue Bloods includes one of my favorite episodes in which Frank investigates a dead priest who is up for canonization. The priest deeply loved a woman in his parish, stating, "He was grateful to God for the challenge. He just wished the challenge didn't have such a great pair of legs."
Jokes rely on a common denominator--otherwise, they aren't funny. Likewise, priests as romantic partners rely on a common denominator. Even in my futuristic world, I rely on "here's what priests are not anymore."
What will happen when/if American television doesn't even have Catholicism as its backup? Maybe every episode will have to come with footnotes.