I want to develop an app for a website (let's say stackoverflow.com in this case).
Will I violate the Blackberry terms of service if I list an (paid) app for a website which I do not own? What about the website I'm going to use?
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If you're actually developing an SO app and not just using it as an example, you can use the content so long as you attribute it to the original creator. All Stack Overflow content is licensed under CC-BY-SA.
This doesn't apply in general. Some sites will have similar licensing for their content, some will be entirely proprietary. You should always check with the site to find out the appropriate licensing or to work out a business arrangement for content access.
I can't find the legal terms for developers of Blackberry applications, but I think it's safe to assume you would not be allowed to publish an application to the Blackberry App World without having appropriate rights to use another site's content. I'm not sure what would happen if you just distributed the application yourself.
And as with all legal matters... if in doubt, talk to a lawyer.