"Who" versus "whom" with multiple verbs

"I know the man to be unhappy" seems be more informative, but it doesn't answer the question until you change it again to "Him I know to be unhappy". Or how about "I know that man [him] to be unhappy. This would indicate that you want the objective case - whom.

As the question is tagged with acceptability, I will report the following paragraph, reported by NOAD in the usage of who section:

The normal practice in modern English is to use who instead of whom (Who do you think we should support?) and, where applicable, to put the preposition at the end of the sentence (Who do you wish to speak to?). Such uses are today broadly accepted in standard English, but in formal writing it is best to maintain the distinction.

If you want to avoid writing who when you should use whom (or vice versa), you can use that.

the man that I know to be unhappy

That is a relative pronoun used to introduce a defining or restrictive clause, especially one essential to identification; it is used instead of when, which, who, whom.

the book that I have bought yesterday
the person that I will meet tomorrow
the year that Anna was born

To answer the direct question ...

Verb clauses can be nested and are parsed from the inner most to the outermost. so "I know 'x to be" is a verb clause which after being parsed would resolve effectively to "is" and therefore your second statement is true - it the correct stereotype verb clause would be "the man who is unhappy".

The response to the other comment - I don't know if this a culture thing but I do not find it to be correct usage to replace who with that. That can only be used when talking about NOT-people.

The quantity of verbs has no effect on the choice between who and whom. The only thing you need to figure out is whether or not you need a subject for a verb.

If who/whom is the subject of a verb, use who. Otherwise, use whom.

The non-technical instruction on choosing the right word: Who can only be used as a subject, so if you don’t need a subject, don’t use who; use whom.

For this example, the correct choice is whom. “The man who/whom I know to be unhappy…” I suppose this is followed by a verb. The simple subject of the sentence (of the verb that follows) is man, not who/whom. Who/whom is not the subject of a verb, so you use whom.

Another way to write the example, which might make it easier to parse, is as follows: “The man, whom I know is unhappy,…” As we see, whom is not serving as the subject of any verb.

Source: Precise Edit

The main verb in your question is "know", so it is "The man whom I know to be unhappy", just as it is "The man whom I know".

It gets more complex if you replace 'to be' with 'is', as there are several possible meanings. "The man who, I know, is unhappy" is equivalent to "The man who is unhappy (I know it)", so whom would be wrong. "The man, whom I know, is unhappy" = "The man is unhappy: I know him (not he). Without any commas, or (just as wrong) with a single comma after 'know', ambiguity makes it impossible to say what the pronoun should be (unless the rest of the sentence makes it clear). Moral: punctuation is important, and don't lazily cut "to be" down to "is" unless you are clear about how you are changing the meaning.

NB Precise Edit's answer (quoted by Lauren), leaves out all the commas in this phrase, so isn't helpful.

As I write, all other answers agree that whom is the correct choice in this construction (because "whom I know to be unhappy" is an auxiliary phrase, wherein whom is not the subject of a verb).

I don't dispute the strict grammatical position, but I would say that, as suggested by this NGram, whom appears to be increasingly falling into disuse.

Correspondingly, here are over 1000 written instances of "who I know to", most if not all of which are "incorrect" according to strict grammar. In my opinion, whom is already becoming somewhat 'dated', and it's only a matter of time before it disappears completely.

The man who I know is unhappy. 'is unhappy' is a clause and the object of the verb 'know' every clause needs a subject. The verb 'is' in the clause needs a subject and every subject must be in the subjective case. So, the subjects in this sentence are I, the man or he, and who. 'is' needs a subject and 'who' should take the place of the man or he. Whom is wrong and can not be subject of 'is'. To solve the problems in this sentence, every verb in the sentence must have a subject. So, 'who' is the perfect subject for the linking verb 'is'.

Very simple. MAN is the subject of the main clause. The VERB of the main clause is IS. (This man is unhappy). "who I know" is a subordinate adjective clause modifying MAN. The person speaking knows the man. The subject of the subordinate clause is I. The verb is KNOW. WHOM is what we call the subordinate clause marker, but it is also the presumed object of the verb KNOW, so it takes the objective (Whom) case, not the subjective. When pronouns follow a TO BE verb, and refer back to the subject of the sentence, they take the SUBJECTIVE case. (I am HE. She is WHO?) Most of the time, one's ear tells one what word to use.

We say
'The man who is unhappy'
But,
'The man whom I know to be unhappy'

Because in the first example 'is' is the verb, in the second 'know' is the verb. In the first, 'the man' is the subject to 'is'. In the second, 'the man' is the object to 'know'.