I have created a test file containing NULL bytes. This file may be served as plain text or as HTML.
Viewing the text file in Firefox replaces the NULL bytes with the replacement character. (Related to bug 451557?). Select All, Copy on this file copies the replacement characters. For reference, Chrome ignores the NULL bytes completely, and does not display nor copy them at all.
Viewing the source of the HTML file in Firefox shows the \x00 byte icon. Doing a Select All, Copy copies the NULL bytes. Again, Chrome ignores the NULL bytes completely.
Pasting the result of Firefox's HTML source view into a Firefox textarea fails: the pasted text is cut off at the first NULL byte. The text after the NULL bytes is missing. It seems that any other behavior (including the NULL bytes, including the replacement character, skipping the NULL bytes completely) would be preferable. When pasting the result of Firefox's HTML source view into Chrome, the NULL bytes are included as NULL bytes.
It seems like the main problem is the pasting of a string containing NULL bytes into a Firefox textarea, but I'm also not sure what the desired behavior is for viewing such bytes in the different types of files.