On Friday, April 13, 2012, Square Enix responded to my question asking them about Machinima. I was in works on creating a youtube series for people to enjoy. I had written a script, had people ready to voice act, etc. Square Enix basically stated that they prohibit the use of anything from their products. The email basically stated that if they wanted to do it, they would hire one of their partners to do so. So to answer your question, they do not allow Machinima. Looking into it further by law, many creators use the "fair use" law which basically states that it's not an exact duplicate copy of game. It all depends on the copyright information. Here's is Square Enix's copyright info.

All copyrighted and copyrightable materials on this Site, including, without limitation, the Square Enix logo, designs, text, graphics, pictures, software, sound files and other files, and the selection and arrangement thereof ("Materials") are © the Square Enix group of companies and/or its licensors. All rights reserved. Except as stated herein, none of the Materials may be copied, reproduced, distributed, republished, downloaded, displayed, posted or transmitted in any form or by any means, including, but not limited to, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of Square Enix or the respective copyright owner. Permission is granted to display, copy, distribute and download the Materials on this Site for your personal use, non-commercial use, and informational use only; provided that, you may not, without the permission of Square Enix or the respective copyright owner, (a) copy, publish, or post any Materials on any computer network or broadcast or publications media, (b) modify the Materials, or (c) remove or alter any copyright and other proprietary notices contained in the Materials.

Basically, any fan art, fan videos, ect that are used are subject to copyright infringement or, in other words, do it at your own risk.