I've commented out my entire ~/.profile and still get this permission denied error when I try to source it.

ls -l gives

-rwxrwxrwx. 1 <myname> users 288 May 21 09:20 /home/<myname>/.profile

groups shows I'm in the users group. Why can't I source my profile?

May be I am misunderstanding here? But, you need to use:

So, add the word source before ~/.profile

But what does "ls -dl /usr/bin" return?
It should give an output something like this:
that is: readable and searchable (x) for everyone.

ls -dl /usr/bin gives

dr-xr-xr-x. 2 root root 86016 Apr 24 09:58 /usr/bin

@dc.901 I've tried both source ~/.profile and . ~/.profile.

By "commented out" do you mean that you've commented every line in the .profile script? Why? To try to eliminate the error?
Is there a line containing ?? Note the double '/' (Which shouldn't matter)
Please paste the error message between [code] [/code] tags, which will preserve exactly what it says. Please use [code] tags for all pasted output, for clarity.
Whats in /etc/profile?

Probably not a good idea to have ~/.profile be group and world writable (chmod 777) Should only be writable by the owner (chmod 644), and need not be executable at all to be sourced.

I'm running CentOS 7 and using bash...I don't even have a .profile -- bash uses .bash_profile

Yes, I commented out every line to remove the possibility of the issue being something within the file.

The error in code tags is

I had 777'd it to see if that would fix the permission error.

Well, the permission error is not reporting .profile, and it didn't fix it, so you should change that back to what it was.
You should put the lines back too, since that didn't fix it either.

That path is very strange. The double / shouldn't matter, but where's the dot (.) coming from?
Where is /usr/bin<anything> used in .profile?
What shell are you using?
Again, what's in /etc/profile? Have you modified it? Or any of the files in /etc/profile.d?

Nowhere, which is why this error is so weird to me.

echo $0 gives

So I guess I should be using ~/.tcshrc instead of .profile. Though I thought I should be able to source any text file.

/etc/profile is a large script file which hasn't been edited, which is installed with the OS, isn't it?

Yes, and if it hasn't been edited that's good. I think it's sourced in .profile -- maybe -- as I said, I don't have a .profile

Is the error causing any problems?
Can you run commands that are in /usr/bin ?
Do you get the error when you first login, or only when trying to source the file?

Well I guess it's not causing problems anymore now that I realized I should be editing the .tcshrc file.

csh/tcsh is much different from sh/bash/ksh/zsh.
The latter read .profile at a system login.
csh/tcsh read .cshrc (tcsh takes .tcshrc instead if it exists), and then .login if it is a login shell (at a system login).