Why does having <TAB> mapped affect <C-I>? The short answer is, "historical reasons", dating from even before the original 'vi'.
The ASCII code for <TAB> is 9, same as <CTRL-I>. Since terminals receive their input encoded in ASCII, they can't tell whether that "TAB" signal came from the actual <TAB> key, or from the user holding CTRL and pressing I. Since Vim was originally written to run on terminals, it can't tell the difference either.
A couple of other pairs of indistinguishable keys are <C-M> with <Return>, and <C-[> with <Esc>.
It's possible there's some arcane way to tell the difference between the two (more likely if you're using GVim), but if there is, I don't know it. As a workaround, you could use nnoremap <SomeOtherKey> <C-I> to give <C-I>'s original function to some other key.