I'm creating new site in Drupal 7, and I must use about 400 panel pages with Views. I'm wondering how many panel pages I can use in my site without affecting performance? Can Drupal/Panels handle 400 panel pages?
There is no reason to believe that "Panels is slow". This is a misconception created by the fact that Page manager and Panels modules are complicated, and not easily understood.
Panels is actually sometimes faster than core is on it's own. And that's without utilizing caching, something Panels can do much better than core, because it allows you to do it from on a page by page basis, down to on a block by block basis.
At NodeOne, we build all our sites using Page manager, Panels, and Panels Everywhere. I have yet to see a performance problem caused by the use of Panels itself. That said, you should consider using Caching, as a website utilizing Panels well is most likely a complex one. (I'm no longer with that business, but this remained true for as long as I was)