Romance can be a great addition to your game. How many action/fantasy/horror/genre movies or TV shows or novels have you watched/read that do NOT have some kind of romantic plot threads? Perhaps a handful out of thousands. Everything from The Iliad to Game of Thrones is driven in large part by romance/love/sex. You may have heard the term "sex and violence," right, because both of those are key parts of most dramatic storytelling.
If someone just wants to "bang bar wenches" then it's fine to say "whatever, sure, back to the game." But it seems to me that in most games GMs are desperate to get PCs to give a single good Goddamn about any NPC and be motivated to do anything for or with them. When they hand you potential romance they are handing you one of the biggest gifts they can - investment in the fictional world.
How To Do It?
First, you need people to be "comfortable" with the topic. It's frankly bizarre that a gaming group can sit down and watch an episode of Deadwood and then walk to the gaming table and then when a PC flirts with someone they get all "tee hee" like they're 10 year old boys. But this is the key issue to deal with. Discuss it with the group and say "Hey, we should be able to do this, so expect it." I had that discussion with a GM who was uncomfortable with RPing with me as my character showed interest in an NPC. This goes beyond the usual suspected causes - in this case my GM was gay and I wasn't, his discomfort wasn't about "yuck I am flirting with a dude", it was just that same tentativeness you and others are feeling, and we had an open discussion about being able to do it and how to do it well. It's not about coming up with hard lines as to where you "shut off the camera," it's more about GM and PCs continually being sensitive to what's interesting as part of the story being told.
Frankly, from an immersion perspective you should be keeping a strong differentiation between in character and out of character talk and action at all times anyway. If you are doing that, then a lot of the confusion about whether it's the person or the character doing the flirting should be dispelled automatically.
Consider trying some storytelling type games like Fiasco, or even one or several members of the group going to an improv acting workshop (there's guaranteed to be one somewhere near you). The main goal here is just being able to man (or woman) up and freaking act in character without coming apart at the seams.
Making It Interesting
OK, so you don't want to spend table time on things the group doesn't find interesting. However, the group there is audience as well as participants. Not everyone has to have their character engaged in every scene to be interested.
Go read David Mamet's note on how to do screenwriting to get some insight into what to put into a scene that's "on camera," so to speak, it'll help you more than a single answer here can.
OK, now that you're back, in my current campaign one PC (ironically, the GM from above who had been uncomfortable with this to start) decided to date and then marry a NPC. Rather than the other PCs being "bored with" this, he and I both took steps to ensure they were engaged. Some of that is just making the RP interesting and dramatic to where it's entertaining to watch. Some of it is making it very plot related (she went from being just a random related NPC to being a lot more integral to the plot once he brought the spotlight to her). Also, he would engage with the other PCs about it just like a real person would with his friends - they became invested in his success in the relationship, would give him advice (his character is a low-CHA barbarian/ranger/druid Viking, not one for the smooth talk), took joy in skedaddling and leaving him to his own devices when she initiated a "define the relationship" talk, started bugging him about when they were going to have kids... There's not one single trick, it's about organically making the NPC and the relationship an interesting part of what's going on.
In this case she was a half-elf, she got kidnapped by the bad guys and he saved her and they stared dating, but then she was revealed to be a serpentfolk (like the BBEG), but he declared his love for her anyway and they got married, and then later decided they wanted to have kids and were on the look-out for any magical aid they could get in that vein. Now they're all sailing around on a pirate ship committing acts of ultraviolence waiting for their egg to hatch. Heck, just the shopping trip to get a custom sea chest to keep the egg safe engaged the other PCs. "Enchant it with alarm! And make sure it floats!"
- It is fine to have romance in your game.
- It's pretty simple to have a discussion with your group about "hey, in character it's fine to flirt and have romances and stuff. We all good?"
- Keep it interesting - maybe that's its own drama, maybe it's being part of the rest of the plot, maybe it's a mix.
- Use it to forward the plot, that's what it's there for.