Serious Eats - seriouseats.com

While Starbucks or another corporate coffee chain might serve the ice without flinching, he doesn't want Murky Coffee's top-notch beans to taste like a "ghetto latte."

The problem with the ghetto latte from the owner's view is not taste -- it's cost. It means the customer is saving money by not paying for milk and instead filling his/her cup with free milk or cream from the bar.

Someone said maybe the shop doesn't sell iced drinks -- not the case, if you read the full story you'll see that they offer other espresso drinks iced including iced Americanos. The difference is that this customer didn't want the extra water. Hence, leaving space in the cup and allowing him to make a ghetto latte (which he apparently had no intention of doing). I suspect this is the main reason the shop has this policy and not because they are worried about the taste.