I ask because I don’t really have any experience with native apps.
With web apps, I’d argue that nothing should be sliced images. CSS is so powerful these days that it really should be able to handle most designs thrown at it. Having said that, the designer needs to be aware of the limitations of the web. It goes both ways.
You’re definitely going to end up with small little icon images, but gone are the days of splicing out an entire header. These days, you can even use icon fonts instead of images.