I've used the Foursquare API for a lot of different kinds of applications, both web and mobile. In my opinion you should not store any venue information in your application for a couple of reasons.

First, I think the terms and conditions you agreed to when signing up for an API key prohibit you from doing so. Second, That data is subject to change at any moment so you're better off just referring to them to get it whenever you need it.

If you have any other questions about the Foursquare API I'd be glad to talk to you and see if I can help.

Thanks