To clarify, the screen only shakes when I try to select text that I typed in a reply box or whatever you want call it. For example, if I didn't like a large section of something I typed I'll I select it while scrolling up or down the page at the same time and the screen shakes like crazy.
It's done this on my Galaxy Nexus, Note 3, and now my Note 4.
If I can find a good screen recording app I'll record it and post it here.
Edit- here's a video: