How can I restrict the use of other applications during an online exam?

There is a very simple solution: Set a time limit. Any student who has to browse the web for an answer will lose a lot of time compared to the ones who know the answer immediately.

Another solution: Don't care. It's not your job to make sure that students don't know about the Internet. Your job is to make sure they know their stuff when they start a job. So if a student is not able to memorize something but can look it up on Google? Why not? I'd award her extra points for working smarter rather than harder.

Speaking as a browser user, I hope it's not possible, since it implies that any web site could do the same thing, thus taking control of my computer away from me, against my will.

It is possible, but only through workarounds.

You simply cannot limit the features of the operating system and the running application as unwind said.

But you can implement restrictions by adding an Ajax framework. You can set it to send the answer whenever focus is lost on the window of the examination, thus forcing your students be honest. This also has many downsides, but this is the one way of implementation I can imagine.

Update

In order to reply to comments, I have rethought and improved the idea. Whenever focus is stolen, one has to enter a captcha. This discourages changing browser windows for every single question, and is not really frustrating if happens once in a while.