Would it be possible to put a TI-86 OS on a TI-84 (or vise versa)

Yes.
http://www.detachedsolutions.com/emu8x/

I can't get it to work, when I type "java rom" it says java isn't a recognized command, help.

Wait a sec - this seems unreal. I actually can emulate an 86 (or an 85 in my case) on my 84+SE? I am surprised I've never heard of it!

Does the 84+ have enough RAM pages (128K's worth)? Is the paging hardware the same? Surely some minor modifications would be needed to the ROM code, at the very least, though I could be wrong.

And what about screen size and LCD controller-vs.-memory-mapped interface differences?

Unless you're just asking "can I store an 86 ROM on my 84+?", not "can I store an 86 ROM on my 84+ and actually run it?".

The oldest 84+(SE) hardware revisions have 128 KB of RAM, recent revisions have only 48 KB. The 84+CSE has 128 KB.

Please engage your brain or use Google before posting simple questions.

I think that Emu8x 86 will only run on TI-84 Plus and Silver Edition calcs made before April 2007. The newer ones have a total of 48 KB of RAM instead of 128, as Lionel says, so TI-86 emulation will be a problem.