A suicide bomber blew himself off, or blew himself up in a mosque?

Solution 1:

To "blow oneself up" is grammatically correct in this context, and is definitely what the announcer should have said. "Blow up" idiomatically means "explode". (It's also used to mean "inflate" in the case of e.g. balloons.)

"Blow off" is even more colloquial; the usage I know means to ignore someone/something - to not attend a class, or show up to a meeting with someone, for example.

Solution 2:

A bomber blows oneself up.

The form of blows/blew is used depending on which tense is used:

Blew himself up. Past tense

Blows himself up. Present tense.

Blow/blew "off", means to ignore or miss an appointment with someone.