It seems to depend on the book - I've picked up a bunch of the Project Gutenburg books and the copy function works on them. But, they're the free ones! On books I've purchased, the copy function isn't there, presumably blocked for copyright protection.
The Kindle app won't let you copy either. But I just checked and NPR News and BBC news both allow copying - USA Today doesn't. If you can copy something, you can simply paste into an email to yourself to transfer it somewhere else.
I guess we'll have to go through each app and see if it will copy. I suspect DRM-protected books will always have 'copy' disabled.
hope this helps! Scott