99.99% of the time, that wouldn't sound natural. The only time I can think of using that sentence would be introducing the months on a calendar to a five-year-old.
"It" is the "dummy subject" used when indicating time or weather. "It's been colder than -20°C all month."
"It" and "that" can be used interchangeably as "dummy subjects" for any subject that is ambiguous or not introduced until later in the clause/sentence, "It/That was very kind of you," (ambiguous subject).
"That" works fine as a "dummy subject" for almost anything aside from time/weather—unless the speaker is pointing at (or otherwise indicating) whatever they're talking about, e.g., a clock (The man in the bowler hat growls at the platform attendant, "The five o'clock train isn't here," and jabbing his finger at his gold watch, he adds, "and that is five o'clock!"), a calendar (insert Luke's example), or a photograph of a twilight scene (Is that morning or evening?).
"There" is the third (and least common) "dummy subject". "There" is used to indicate that someone/something (allegedly) exists/existed in a location, e.g., "There were three trolls in the forest arguing about how to eat me!"
"Det er ... ," replaces all three English "dummy subjects" ... except when referring to time (Klokka er tre over fem!)