Punctuation marks, like words and paragraph breaks, are tools. Overuse of any tool will make your writing inelegant, but using the proper tool at the right time will help you generate pages that are well crafted and precisely assembled.
In fiction, as in other kinds of writing, you'll still want to use the em dash to indicate interruptions, performing a function related to — but subtly different from — parentheses. (The em dash is particularly useful in dialogue.)
There is a distressing tendency for some writers to overuse the em dash, and I often notice this kind of overuse in amateurish fiction where there are also other problems. Overuse of the em dash (or parentheses, or the semicolon — or any punctuation marks) can make your prose a little predictable. The em dash also can make prose seem disjointed and stuttery, almost as if you put little thought into your writing — putting phrases that should come earlier at the end of a sentence, for example.
Em dashes are easily edited out, and the clumsy sentences rewritten. Every writers' habits are different, of course, but getting yourself into a habit of using dashes sparingly may well train you to learn to think in terms of variety.
I'd reserve the em dash for situations where you want to startle the reader — startle them just a little bit. Used occasionally, this kind of construction can be a useful tool that will add variety to your prose. You can also do this with, say, parenthetical phrases, or by varying average sentence or paragraph length. But doing any of these things will make your writing stale and predictable, so treat them as unique hand tools you break out when the power drill/screwdriver won't work elegantly, and you need just the right tool for the job.