It "bursts into flames" and then the message is relayed anyway.
We see a howler explode in Harry Potter and the Order of the Phoenix when Petunia takes too long to open the Howler Dumbledore sent her:
"You can open it if you like," said Harry, "but I’ll hear what it says anyway. That's a Howler."
Aunt Petunia's hand was trembling. She looked wildly around the kitchen as though looking for an escape route, but too late -the envelope burst into flames. Aunt Petunia screamed and dropped it.
An awful voice filled the kitchen, echoing in the confined space, issuing from the burning letter on the table.
Remember my last, Petunia.
Harry Potter and the Order of the Phoenix: Chapter 2 - "A Peck of Owls"
This is also confirmed by Percy in Harry Potter and the Goblet of Fire:
"It's been absolute uproar," Percy told them importantly, the Sunday evening before they were due to return to Hogwarts. "I've been putting out fires all week. People keep sending Howlers and of course, if you don't open a Howler straight away, it explodes. Scorch marks all over my desk and my best quill reduced to cinders."
Harry Potter and the Goblet of Fire: Chapter 10 - "Mayhem at the Ministry"
And again when Hermione gets some hate mail after Rita Skeeter's article about her was written:
Hate mail continued to arrive for Hermione over the following week, and although she followed Hagrid's advice and stopped opening it, several of her ill-wishers sent Howlers, which exploded at the Gryffindor table and shrieked insults at her for the whole Hall to hear.
Harry Potter and the Goblet of Fire: Chapter 28 - "The Madness of Mr Crouch"