Is 'to resign' an object or subject complement in 'The teacher wishes to resign'?

In '[The teacher] wishes [to resign]', '[to resign]' can be analysed as the subject complement because it is ascribing a property related to that of the subject noun 'The teacher' and hence, modifying it.

This construction is widely recognized as a "chain of verbs" (catenative, according to the CGEL terminology). This said, "to resign" is a to-infinitival clause, an extremely common form of a verb complement. The head of the clause is the verb "resign". The subject of this clause is the same as that of the matrix clause. (the teacher wishes, the teacher resigns). Were the subjects different, the understood subject of the to-infinitival would be aligned with the object of the main clause: The teacher wishes me to resign. (The teacher wishes, I resign).

Most importantly, clauses do not "ascribe properties" to anything. We think of clauses in different terms. Subject and object can be ascribed properties in the form of an adjective, noun or (less systematically) prepositional phrase.

EDIT: cross-posted with TaliesinMerlin

TaliesinMerlin already provided an accurate (or at least very widely accepted) analysis of this construction, so I could as well delete my post :). However, I could add a couple points to supplement his post.

The examples from Grammaring illustrate the use of to-infinitival clauses as subject complement, but they are possible in this position only as specifiers of the subject. Clauses, as I said, cannot "ascribe properties" to anything. This means that it might be a bit inaccurate to say that "to infinitivals" in these examples "elaborate" on the subject. They can only specify the subject: What is the advice? What is the decision? What is the thing that is essential? (CGEL p270 says that "In the ascriptive construction a subordinate clause can function as S but not in general as PC" In the footnote they note that this may be possible to a limited degree, such as with purpose clauses: This is to clean the lens with etc)

Adjectives and noun phrases could be said to "elaborate on the subject". Adjectives, by definition, ascribe properties, while noun phrases may be used either way.

Are you looking for CGEL terminology? In your sentence—

The teacher wishes to resign.

—to resign is a catenative complement. It complements the catenative verb wishes.

Here is what CGEL has to say in 14 1.2:

[13] vi a. She wants _to leave the country_.
        b. She seems _to like them_.
        c. She hopes _to hear from them soon._

. . . we analyse the underlined clauses in [13vi] as examples of a distinct type of complement realised exclusively by non-finite clauses; we refer to them as catenative complements. . . . The term ‘catenative’ applies to a large class of constructions where a verb has a non-finite internal complement. The name reflects the fact that the construction can be repeated recursively, yielding a concatenation (‘chain’) of verbs . . .

Some traditional grammars consider each non-finite verb in the "chain" after the main verb to be the direct object of the preceding verb, which means it functions as a noun, which means it will be either a gerund or a noun infinitive.

The Guide to Grammar and Writing states:

. . . both gerunds and infinitives can act as a Direct Object . . . Verbs that take other verb forms as objects are called catenatives (from a word that means to link, as in a chain). Catenatives can be found at the head of a series of linked constructions, as in "We agreed _to try to decide to stop eating_ between meals."

(The site goes on to offer a handy list of verbs that accept non-finite verbs as direct objects.)

Consider, for example, this chain:

The teacher wishes to consider planning to resign.

Wishes what? To consider (noun infinitive). Consider what? Planning (gerund). Planning what? To resign (noun infinitive).

Resign is not a catenative verb—it can't be followed by a non-finite verb—thus ends the chain.

Further reading: Catenative Verbs