I can confirm an issue with the 404 page. When running PHP 5.6.24 and type a random URL I get a 404 - Article not found error.
When running PHP 7.0.9 I get a 0 - Call to a member function get() on null error.

I tested this on latest staging and Joomla! 3.6 stable with the error reporting on Develop. On another Joomla! 3.6 stable website running PHP 7.0.8 there is no problem. Not sure if this is an other issue, because I don't get a 403 error

Georges-MacBook-Pro:jed3-dev george$ curl http://localhost/~george/joomla-cms/?option=com_fooooo -I
HTTP/1.1 404 Component not found.
Date: Sat, 30 Jul 2016 11:03:33 GMT
Server: Apache/2.4.18 (Unix) PHP/7.0.9
X-Powered-By: PHP/7.0.9
Set-Cookie: 145fa16e3cae7251293782cff47cfb8d=c8et1ipfanhgt7bp67d9snc016; path=/; HttpOnly
Cache-Control: no-cache
Pragma: no-cache
Content-Type: text/html; charset=UTF-8

I can't confirm such an issue on staging with PHP 7.0.9 :(

Database Version    5.5.5-10.1.14-MariaDB
Database Collation  utf8_general_ci
Database Connection Collation   utf8mb4_general_ci
PHP Version 7.0.9
Web Server  Apache/2.4.18 (Unix) PHP/7.0.9
WebServer to PHP Interface  apache2handler
Joomla! Version Joomla! 3.6.1-rc2-dev Development [ Noether ] 30-July-2016 11:14 GMT
Joomla! Platform Version    Joomla Platform 13.1.0 Stable [ Curiosity ] 24-Apr-2013 00:00 GMT
User Agent  Mozilla/5.0 (Macintosh; Intel Mac OS X 10_11_6) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/51.0.2704.103 Safari/537.36

I see the original report was on nginx and not Apache. Perhaps that's the
issue?

I can confirm multiple issues with Joomla 3.6.0 and 3.6.1rc1 on php7.0.9

Maybe. Generating a 404 a different way (how I got a 403 on your test site @brianteeman ) I got on this local setup

Georges-MacBook-Pro:joomla-cms george$ curl -I http://localhost/~george/joomla-cms/foobar
HTTP/1.1 500 Call to a member function get() on null
Date: Sat, 30 Jul 2016 11:41:25 GMT
Server: Apache/2.4.18 (Unix) PHP/7.0.9
X-Powered-By: PHP/7.0.9
Set-Cookie: 145fa16e3cae7251293782cff47cfb8d=t97887dhgilni62861em6f0ig6; path=/; HttpOnly
Cache-Control: no-cache
Pragma: no-cache
Connection: close
Content-Type: text/html; charset=UTF-8

@wilsonge your result is correct, but when you type someting like http://joomla-cms.dev/index.php/test I get an 0 - Call to a member function get() on null error

I am closing this as it looks like it is a duplicate of #11360 which has a PR for testing

@brianteeman the issue I have looks like a different issue?

Re-opened - sorry it is not the exact same issue

i think you are wrong and it is the same issue resolved by #11360
http://dev-3-6-1-site-2.brian.deevop.site/ggjgjhgj with patch from #11360 applied all good

http://dev-3-6-0-site-2.brian.deevop.site/ggjgjhgj without patch applied error and 403

Yes you're right, when applying that PR the 0 - Call to a member function get() on null error is gone :)

This is fixed with me merging #11360