Obsess is defined as: (verb) to dominate or preoccupy the thoughts, feelings, or desires of (a person); beset, trouble, or haunt persistently or abnormally.
I've read a lot of things that say you shouldn't fall in love with your writing—that getting overly attached can only lead to heartache. It should come as no surprise that I completely disagree. And not just because I obsess over my own writing. Look at the definition again. Doesn't it describe the writing process perfectly? When you're working on a story, it certainly preoccupies your thoughts, probably your feelings and desires to an extent as well (you desire to write more than anything, right?). And I've certainly been haunted by a story idea before. So obsession is really just a natural occurrence when it comes to writing. And you should embrace it.
I've been an obsessive person since I was at least 14 years old (probably before that, too, I just can't quite remember). I can't help it, really. Usually it's music, Broadway musicals, or (of course) my own writing. I think it's fun. Sure, there are several things that I like, but only a few things that I obsess over. It's a deeper level of liking something. You develop a personal relationship with that certain thing—you know everything about it. If you meet someone who likes it, too, it's like finding a soul mate. And usually, at least for me, obsessions never die. They may fade a bit, but they're always somewhere in the back of your mind waiting to be dug up and obsessed over again.
So why should you obsess over your own writing? Well, it's not really that you should, it's that you're going to. If you're not, then your heart may not be in it. Of course, your level of obsession is up to you—there's mild obsession and insane obsession (I won't say which one I have...). You need to like what you're writing—if there's too much distance between you and your work, then the reader will probably notice. It could seem stale and void of emotion. If you want your characters to have passion, then you should have some passion for them. You want your reader to feel the same attachment to your characters that you felt when you wrote the story.
More than anything, obsession will help you to just get your work done, because you never want to be away from it. So don't fight the obsession—embrace it!