In my campaign right now (my first as a DM) we actually have two NPC-PC relationships going. Its rather ironic, as both NPCs were initially intended as foils.
My first two players (both females), when they picked character classes, decided upon a fighter,and a bard respectively... so they had very little magical firepower (this was back when they weree 5th level). As a way to help, I threw in some NPCs to help... and being a DM that loves to give EVERYTHING backgrounds, I gave the NPCs backgrounds as well.. one was a young noble who was trained with the sword and had magic in his veins (fighter/sorcerer), while the other was his cleric tutor. My intention was to have the noble play a goofy, kid-like sidekick to them while steadily gaining magic to eventually give them a respectively spellcaster in terms of firepower. As my players liked roleplaying, I flipped a coin and decided he'd have a slight, teenage-type crush on the fighter in the party, to give them a chance to flesh out their characters.
Lo and behold, the fighter felt protective over this charge, and then she slowly became attracted to him. Didn't know this till several more sessions, and the roleplaying opportunities it gave her were rewarding (her character trying to decide whether to actually tell him, or not, etc.). About a month in game after they met, she ended up confessing her love to him (they're married now). After that, the roleplaying turned into her protectiveness of him, and his of her, which helped drive several plots.
Meanwhile, my roomate (a male) joined our campaign. To give him some experience before he jumped in, I ran a couple solo sessions for him with his rogue. He gave the character maxed out diplomacy stats and had an 18 charisma, and through some convoluted plans, ended up schmoozing a barmaid with five rolls all above 23 for various items, and assistance in screwing the barowner out of ale. I decided it'd be fun for me as a DM to have her start following him around (basically she was starstruck), and the two hit it off.
Once again, the roleplaying that came out of it was rewarding, as his character suddenly gained depth of opening up to her, letting go of being supremely overconfident and actually becoming human (as well has helping the player figure out a backhistory of his character). At present, these two characters are married, and expecting triplets.
In both cases, things turned out really well and the roleplaying rewards for the players, as well as me, were great. Both of the NPCs, which originally were meant to be either short term or background, ended up being regarded by the players as full equals, and greatly beloved, with personalities and traits (the barmaid has now spent a while studying wizardry under the tutelege of a powerful friend of the party, and now she's a wizard that can beat people over hte heads with beer mugs, for example). Everyone enjoys the sessions immensely when they're around... so much I restructured my campaign to include those two in more adventures.
It depends on your players. Some, like my players, can gain alot out of in character romance and NPCs that bounce off of them. Some campaigns, probably won't. The key is to know your players and their likes and dislikes before proceeding with things. That way, regardless of how they stand on the issue, you can make the campaign be one that they enjoy and have fun in.