Why "the powers that be"?

In the phrase "the powers that be," as in the sentence:

It would never have occurred to the powers that be to run and supervise the National Lottery from anywhere but London.

(Oxford Dictionaries)

Why do we use "be", the infinitive form of the verb? Wouldn't "the powers that are" or "the powers that rule" make a lot more sense?

To expand on Henry's answer: "The powers that be" is a set phrase quoted from Romans 13:1.

Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God.

In that context, it means "the temporal powers that indisputably do exist in the world," so we can rule out explanations that call on the subjunctive mood — here Paul isn't talking about "the powers that [may or may not] be", he's definitely talking about what we today would call "the powers that are."

So, why do archaic English sources use the form be where modern English would use are? Well, here are many partial answers all together in one place: Abbott's A Shakespearian Grammar quotes Richard III, act IV, scene 4:

Where is thy husband now? Where be thy brothers? Where are thy children?

Abbott suggests that Shakespeare (and also the King James translators, who were working in English at the same time as Shakespeare) preferred be to are for the plural indicative form basically because it sounded better on a case-by-case basis; with maybe an additional connotation of uncertainty.

So, as a rule, it will be found that be is used with some notion of doubt, question, thought, &c.; for instance (a) in questions, and (b) after verbs of thinking. [...]

Be is much more common with the plural than the singular. [...]

Be is also used to refer to a number of persons, considered not individually, but as a kind or class. [...]

But it cannot be denied that the desire of euphony or variety seems sometimes the only reason for the use of be [as opposed to] are.

— Abbott, A Shakespearian Grammar, "Auxiliary Verbs", sections 298–300.

Let's face it: "the powers that be" sounds much cooler than "the powers that are", and the guys who wrote the King James Version knew that just as well as you or I. :)

"The powers that be" is a set phrase drawn from early translations of the Bible into English (Tyndale, Geneva, KJV etc.), in particular Romans 13:1.

So its grammar (subjunctive) reflects the usage of the time, and even then might have been slightly archaic.

The phrase "the powers that be" doesn't employ the subjunctive mood. The phrase comes from the New Testament (Romans 13:1) and uses be instead of are as an archaic alternative to the present indicative tense, not as an expression of the present subjunctive mood.

This is explained explicitly in regard to this very phrase in the following Wikipedia article: English subjunctive

This blog says that it's the archaic English subjunctive form. Which I guess we could translate into modern English as "Whatever powers there may be".

However, I think the in sense that most people use it, they don't mean it in the subjective sense (i.e. an expression with doubt), but rather the indicative "the anonymous authorities that are presently in control".