Why make this change?
This is a tricky one. Like “just” or “really,” “that” is often unnecessary. My test is if the sentence is rendered nonsense with “that,” or if the meaning would be changed without it, then keep it. Otherwise, leave it out.
The President said (that) June 1st the new law will take effect.
Without “that,” the President said it on June 1st.
With “that,” the law takes effect June 1st.
You would include/omit in this case depending on what you want the sentence to mean.
Take another example:
There’s no question that “that” is necessary for the sentence to have any meaning in the context of the question.
Now, consider my example:
“This is the beer that I like.”
This is a “that” that should be omitted. Is it technically incorrect? No, I don’t think so. But it’s inartful. The sentence has the same meaning with or without it, and without it the sentence still makes sense. So, leave it out.
The best article I found on the use of “that” is here. The author makes a valid point, which is that using “that” is generally not grammatically incorrect, but omitting it might be.
That’s certainly true, but I submit that peppering your work with unnecessary “thats” is worse than leaving one or two out where they arguably should go. Sometimes it’s a judgment call, sometimes it’s a matter of style. Generally, though, they should be omitted, unless it affects the meaning of the sentence, or renders the sentence nonsense.