You don't have to, but you probably want to for a couple of reasons:
It's courteous, and in the spirit of open source
It's someone else's work, and you're using it. The least you can probably say is "thank you." It will also probably help you stay in the clear: since you're using software in binary form, where the notice isn't immediately accessible, then by providing a copy of the license, you respect that licensing term in another way.
It tells your users what's up with the program
Let's admit it, having the license accessible to the users tells them what's in the program and so on. It's another way of providing attribution, like I listed in the first reason above. Many apps, desktop and mobile, have a screen or panel to indicate the projects and licenses that they use. They don't have to be straight in the user's face, they can be a little button in the "About" screen of the program.
To answer a little confusion: the binaries are still a derived form from that source code. Analyse the heading:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
The software is still there, it's just there in a different form, a compiled form. Therefore, the copyright and permission notice should probably stick around, even if it's just a file somewhere.