The relative pronoun can only be omitted when it is the object of the clause. When the relative pronoun is the subject of the clause, it cannot be omitted. You can usually tell when a relative pronoun is the object of the clause because it is followed by another subject + verb.

When can that be omitted from a noun clause?

Plenty of guidelines are given for the inclusion or omission of that when introducing a noun clause. The recommendations of the AP Style Guide are often quoted: Omit that after the verb to say–“usually.” Do not omit that when a time element intervenes between the the verb and the dependent clause.

Can that be omitted in a sentence?

After a verb of attribution (said, stated, announced, disclosed), the word “that” often can be omitted with no loss of meaning: He said (that) he was tired. No need for "that." Better to omit.