Do writers or producers value the opinion of vocal fans? For instance, if a particular television romance is popular on twitter, do the writers of that show care? What if that romance is hated? Do the opinion of fans matter at all? Or, do television writers simply write what the network/producer tell them to?

Nah they just do as they are told.

I think that the cast/crew of television shows do pay attention to social media. I don't know if the vocal fanbase has any say on plot direction, but I'm sure they're likes and dislikes are heard.

My sister met Ana Ortiz at a fan event in NYC and she told her if "Devious Maids" didn't have a strong online fanbase (live-tweeting) then Lifetime would have canceled it much earlier on.

So, I think social media does have some impact on keeping shows around. I also know that some cast members do live-tweeting w/ fans when their programs air, but I don't know if it's on their own accord or part of their contracts.

I think they get their idea from many forms. On Law and Order some of the crimes on the show actually happened in real life. On Sit coms some of the shows actually happened to the actors or their familes.