Whenever I hear this, this is my reaction:
I would be willing to bet a large amount of money that before you started writing for fun, you had already spent many hours reading for fun. (And if you're a writer who doesn't enjoy reading, just leave. Please, save yourself and the rest of us the trouble. Just go.) You probably starting writing because of your love for reading. It makes no sense to give that up.
It's like that scene from How To Train Your Dragon, where they're standing around the dragon book.
In this scene, Snotlout (Yes: there is a person named Snotlout. And David Tennant voices his dad, Spitelout.) imparts a bit of wisdom on us without even realizing it. Don't tell him; he'd be mortified. Say his line out loud. Sounds dumb, right?
This is what happens to writers who never read. While they probably won't come out sounding like morons, their writing will suffer. You will keep releasing words from your system, but you won't be putting any back in. And while words are not like a waterfall--if you cut off the river, water will stop falling over the edge--your writerly brain still needs word input.
Think about it: how much sense does it make to be a painter who never looks at paintings? A musician or composer who never listens to music? An architect who never admires a building?
Reading has so, so many benefits to the writer. It shows examples of how to make something work. It can give examples of what not to do. Subconsciously, you'll pick up new styles of ways of putting together a sentence. The benefits are endless, and in the end, the only drawback is that you'll have less time for writing. But would you rather have more writing time with a literature-empty mind, or a little less time with a mind brimming with words and sentences and stories? There really is no question.
Reading also gives you a sense of what's out there, in terms of other books. The market is saturated with picture books about children being raised by giant squid, or teenage guys who turn into a moose every half moon? You may want to head another direction.
And besides, it's been scientifically proven that reading makes you more awesome.*
*This fact can be verified by my personal experiences and a large sample of statistical data that exists in my imagination.