Which preposition is correct?

Verbs of speech, like ask, answer, reply, say, or tell, are very complex and can address a number of details, which may or may not correspond with other verbs. Since these are words about words, they have plenty of recursive properties, and quite unique grammars.

Some examples:

Say can take a direct quotation as a direct object, while tell can't

• He said "Arma virumque cano" to her
• *He told her "Arma virumque cano"

Though both say and tell can take that-clause objects

• He said that he was tired (to everybody)
• He told everybody that he was tired

tell requires a dative-moved indirect object, but doesn't require a >direct object at all,

• *He told that he was tired to everybody.
• He told everybody, not just me

say requires a direct object, but not an indirect object, and blocks >dative-movement when there is an indirect object

• *He said (to) everybody, not just me
• *He said (to) everybody that he was tired

As for ask, an indirect object is not required

• He asked (the conductor) "When is the train arriving?"
• He asked (the conductor) when the train was arriving

but if an indirect object is present, it must appear first, by dative movement

• *He asked "When is the train arriving?" (to) the conductor
• *He asked when the train was arriving (to) the conductor

Part of the problem is that to, the ordinary dative preposition, is not correct with ask, since there are two people involved -- one to ask and one to answer, so there are two directions; to simply provides insufficient directional information.

There are several ways around this.

• One can put a question to someone, about or on some topic
• One can question someone, about or on some topic
• One can ask a question of someone, about some topic.

But mostly ask wants its addressee up front, if it isn't already understood in context.
With no preposition at all.

Neither; if you must use ask, then you best bet is:

^?That question should be asked of the teacher, not me.

as suggested in the comments.

Note that to most people, this would sound a bit old-fashioned (as it was also mentioned in the comments; I would say that it is even possible that some might find it downright unacceptable).

However, if you give up on ask, then many possibilities open up:

That question should be posed/directed/addressed/put/presented/submitted to the teacher, not me.

That question should be directed at the teacher, not me.

Discussion

As John Lawler explained, ask prefers that the identity of the entity to whom we are posing the question be supplied as the indirect object. So your sentence in its 'base form' (i.e. in active voice) would (ideally) be

[1] One should ask [the teacher] [that question], not me.

Here the teacher functions as an indirect object, while that question functions as a direct object. Verbs that have both kinds of objects are called ditransitive verbs; those that have just the direct object are called monotransitive verbs. Many verbs can function both monotransitively and ditransitively. Indeed, ask is such a verb.

Moreover, in the case of ask, both objects in [1] can function as the direct object in a monotransitive construction: one should ask [the teacher]; one should ask [that question].

Now, what you want to do is turn that question into the subject of the sentence. In [1], that question is an internal complement of the verb; in a sentence in which it is the subject, it would be an external complement of the verb. Thus, we say that what you wish to do is externalize that question, i.e. turn it from an internal complement of the verb to an external one.

Typically, we externalize by using the passive voice.

The problem is the ditransitive nature of the verb in [1].

A clause in which the verb is ditransitive (i.e. has both an indirect and direct object) can, in principle, have two passives, depending on which of the two objects is externalized. However, for many ditransitive verbs, only the indirect object can be externalized, especially in American English. Here is what CGEL says about that (p. 1432):

Indirect and direct object in ditransitive clauses

[19]  i  a.  My father gave me this watch.     b.  I was given this watch by my father.
       ii  a.  My father gave me this watch.    b.  ^%This watch was given me by my father.

In principle, ditransitive actives have two passive counterparts depending on whether it is the indirect or the direct object that is externalised, as in [ib] and [iib] respectively. The version with the indirect object externalised is called the first passive, while the one with the direct object externalised is the second passive - terms based on the linear position of the relevant object in the active construction. The first passive is much more common; the second is hardly possible in AmE, and even in BrE is acceptable in only a limited range of cases.

The degree to which one or the other passive is acceptable is related to the preposition that is used in the prepositional alternant. Namely, as CGEL says (p. 248), 'most ditransitive clauses have alternants with a single object and a PP complement with to or for as head.' Examples:

I sent Sue a copy. -> I sent a copy to Sue.
I ordered Sue a copy. -> I ordered a copy for Sue.

Here is CGEL (p. 249):

Ditransitive verbs vary considerably in how readily they occur in passive clauses. In general, where Oⁱ [the indirect object] corresponds to the complement of to in the prepositional alternant, characteristically having the semantic role of recipient, the first passive is fully acceptable, as in [I sent Sue a copy -> Sue was sent a copy]; judgements vary as to the acceptability of second passives like [^?A copy was sent Sue]: many find them unacceptable,²⁴ and they are textually quiter are. Where Oⁱ corresponds to the complement of for in [I ordered a copy for Sue], with the semantic role of beneficiary, neither passive is completely acceptable, but many speakers find the first marginally possible [^?Sue was ordered a copy].

^{24Acceptability is greater when we have a personal pronoun + by phrase: This copy was given me by my grandfather.}

Looking at [1], we see that the second passive is perfectly fine:

[2] The teacher should be asked that question, not me.

But the first passive is not:

[3] ^*That question should be asked the teacher, not me.

On the other hand, if the information about who is being asked is left out, there is no problem, for ask does allow the monotransitive construction where the direct object is what is being asked—as long as the entity to whom the question is directed is not stated:

[4] One should ask that question. -> That question should be asked.

Given all that, consider the following. If the passivization of the ditransitive construction is problematic (as it is for [1]), then one may try to work directly with the prepositional alternant. In that case, what used to be the indirect object in a ditransitive verb turns into a direct object in a monotransitive one, and then it can be externalized through passivization without difficulty. For example, the second passive is impossible for I ordered Sue a copy [it would be ^*A copy was ordered Sue]. But if we work with the prepositional alternant, we can successfully externalize a copy, like the this: I ordered a copy for Sue. -> A copy was ordered for Sue.

Thus we have much incentive to look for a prepositional alternant for [1].

Unfortunately, in the case of ask, we run into a little bit of trouble if we try to use a preposition phrase (PP) to add the information about who is being asked; as John Lawler said, ask does not really work well with that type of construction.

As was pointed out in the comments, the best bet is of, but it sounds old-fashioned:

[5] ^{old-fashioned}One should ask that question of the teacher, not me.

Nevertheless, the construction does appear in Corpus of Contemporary American English (COCA):

Perhaps the editors should ask that question of the 200,000 policemen who use the Glock as their duty sidearm.

If we decide that [5] is acceptable, then we can indeed passivize:

[6] That question should be asked of the teacher, not me.