Is there a word or phrase for someone whose ominous remarks come true?

Jinx

TFD n. 1. A person or thing that is believed to bring bad luck.
2. A condition or period of bad luck that appears to have been caused by a specific person or thing.
tr.v. jinxed, jinx·ing, jinx·es
To bring bad luck to.

If a person is jinxed (adjective), it means that he or she always has bad luck. If you are experiencing a series of misfortunes, you can say that somebody has put a jinx on you.

In the OP's examples the person speaking is "responsible" for causing the unfortunate incidents, hence she/he is a jinx (noun).

Vocabulary.com explains it well

If your sister gets a flat tire every single time you ride in her car, she may have good reason to accuse you of being a jinx, or a bringer of bad luck.

When your bad luck makes you think there's a jinx on you, you believe that you're cursed. A bad omen — like a black cat — can be called a jinx, or the resulting bad fortune itself might be referred to that way. Jinx is also a verb, meaning "to subject to bad luck." In the 17th century the word was jyng, which also meant a bird called a "wryneck," used in witchcraft.

The person can be called a Cassandra

(as noun a Cassandra) A prophet of disaster, especially one who is disregarded.

This is based on the character Cassandra from Greek mythology

A daughter of the Trojan king Priam, who was given the gift of prophecy by Apollo. When she cheated him, however, he turned this into a curse by causing her prophecies, though true, to be disbelieved.

[both from Oxford Dictionaries Online]

I'd call that person bad luck

"You're bad luck, you are." _{^{You're my Henry Allbones - G. B. Hope}}

"Maybe you're bad luck." _{^{Hero - E. V. Crowe}}

"You're bad luck. I bet you're bad luck to yourself." "He is, kid," Blocker said. _{^{Yesterday Will Make You Cry - Chester B. Himes}}