In a PHP / Apache / Linux context, why exactly is chmod 777 dangerous?

There are many good general reasons to follow minimalism when it comes to permissions, but in the context of a LAMP webhost, the few that come readily to mind are

• On a shared hosting platform, other users sharing your host can now read and write to your scripts.
• On a dedicated host, rogue processes can read/write and accidentally delete your files. Let's say there is a custom logging process running in the background as user nobody which has a bug that results in it trying to rm -rf /. Now generally this will be harmless because there would hardly be any file that nobody should have write permissions on but this rogue process will now take your files with it.
• To deface your website, someone needs to only gain access as any user, even say nobody or some such dummy account. Generally, the attacker would have to do a further user level escalation attack to get to the place where he can do some damage. This is a real threat. Some non-critical services may be running under dummy accounts and might contain a vulnerability.

It greatly increases the vulnerability profile of your website to malicious activity because it's only necessary to break into one account.

Anyone that gains access to your system with any login can do whatever they want to your pages, including changing them to read "This website is really insecure so please give me your credit card info."

EDIT: (To clarify and address comments)

Many servers have more than one purpose in life. They run multiple services. If you carefully isolate those services from each other by assigning each a unique user and managing file permissions accordingly, yes, you are still in hot water if someone compromises the credentials for an account, but the damage they can do is limited to that one service. If you just have one generic account and set the whole file system to 777, one compromised account jeopardizes everything on the machine.

If your server is dedicated to only running Apache/PHP and serves no other purpose in life, and there is only one account under which Apache/PHP is being run, having that one account compromised is as good as having the whole machine compromised from the point of view of your application (although you should still have system files protected and non-writable by the account used to run PHP... that should still only be possible for an admin account/root).

If they can write a file, and it is executable, they can change it to something that executes on your machine (executable or script) and then use PHP's shell_exec to run that executable. If you're configured not to allow shell_exec, they can change your configuration as well

Here's one scenario:

1. You have an unprotected directory that users can upload to.
2. They upload two files: a shell script, and a php file that has a system() call in it to the shell script.
3. they access the php script they just uploaded by visiting the url in their browser, causing the shell script to execute.

If this directory is 777, that means that anybody (including the user apache, which is what php script will execute as) can execute it! If the execute bit is not set on that directory and presumably the files inside the directory, then step 3 above would do nothing.

edit from the comments: it's not the PHP file's permissions that matter, it's the system() call inside the PHP file that will be executed as a linux system call by the linux user apache (or whatever you have apache set to run as), and that is PRECISELY where the execution bit matters.