Do I need to protect my desktop app if distributed over AppStore?

Mac App Store slips a receipt each time it is downloaded into the bundle. The receipt contains information about the computer used (the so called GUID) and the user logged into the App Store.

See here how you should validate the receipt: https://developer.apple.com/library/mac/releasenotes/General/ValidateAppStoreReceipt/Introduction.html#//apple_ref/doc/uid/TP40010573

If you implement the GUID validation as described in the document, the app will not run on any other computer.

Also check the signature of your code to make sure it has not been tampered: Verifying app's signature by code

Your application, when downloaded from the App Store, contains a receipt. The receipt contains proof that it was downloaded onto this computer from the App Store, and the ID of the application. There are instructions somewhere on Apple's website that tell you how to verify the receipt and what to do if the verification fails.

That said, you are much better off concentrating on writing an app that people actually want to buy. People who pirate your app wouldn't hand over money if it couldn't be pirated. They would do without it, pick some free app, or pick a different app that they can pirate.

And I'm quite sure that any copy protection you build into your app yourself will get it rejected from the App Store.

Even though the answers I've got were helpful, they didn't quite provide the answer I needed. Looking around I found a software called Receigen from Laurent Etiemble, and the FAQ on his site (http://receigen.etiemble.com/faq.html) gave many answers for what I was looking for. For instance

What is an App Store receipt validation ? Why is it needed ? Basically, an App Store receipt is what an application must check to ensure that the copy is genuine and can be run.

What happens if I don't check the App Store receipt ? Well, anybody with a copy of your application can run it, with or without proper authorization.

Is the code receipt validation easy to write? No because it requires deep understanding of cryptography and secure coding techniques.

It didn't hurt either that Receigen generated code that freed me from dealing with receipt validation code. Receigen takes care of this part so I can focus on what is really important for me: my application.

Yes, it costs money but personally I am more than glad to pay it, because I find this part of the development tedious, boring and complex.