He's not being honest. (Para. 9)
Meaning: "He is, for the present time, not saying something honest. He is lying."
"Be" is not usually used as the main verb in progressive tenses. However, you can use it in progressive tenses to describe someone's behavior at a particular time as in the sentence. Here some more examples:
You're being very silly. (= You're doing silly things.) 你做得傻极了。
I was being very careful. (= I was doing something carefully.) 我是特别小心。
He is being nice to me. 他对我显得特别好。