Note: A noun clause is a subordinate clause that answers “what?” after a verb in another clause: “I feel that you are mistaken.” Main clause: “I feel.” Noun clause: “that you are mistaken.”
Most of the time, I agree with their judgment and remove the offending that. Sometimes, however, I choose to leave it in, even if it’s not strictly necessary.
The modern mantra of “leave out needless words” is one to observe in a general way, but it shouldn’t lead a writer to slash mindlessly at every word that can be left out just because it can be.
Plenty of guidelines are given for the inclusion or omission of that when introducing a noun clause. The recommendations of the AP Stylebook are often quoted:
Recognizing the impossibility of laying down hard and fast rules for the use of that as a conjunction, the AP entry concludes with this sensible remark:
When in doubt, include that. Omission can hurt. Inclusion never does.
Fowler mentions some additional verbs that usually require a that: agree, assume, calculate, conceive, hold, learn, maintain, and suggest.