If you had been my child, I would have demanded that such an irresponsible teacher be fired.
It is the subjunctive mood, I suppose. Why every verb in the sentence in the past, but be is in the present?
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Subjunctive tenses are a little more abstract with respect to time than indicative ones are. This is a good overview of usage: http://en.wikipedia.org/wiki/English_subjunctive
With the verb to be, past tense implies a condition known not to be the case, whereas present tense doesn't imply that knowledge. For example:
If Abraham Lincoln were alive today, he probably wouldn't be able to run for president.
Be he alive or be he dead, I'll grind his bones to make my bread.