I was rather surprised to see this tweet, and was even more surprised to see it being retweeted.
So, when do we use 'whom'? To be honest, it is rarely used in informal spoken conversations, and you're more likely to see it in written form, but let's take a more detailed look at 'who' and 'whom'.
As a question word
Who is used without a following noun to ask about people.
Who is that boy standing over there?
Who was your favourite teacher?
Who said that?
Note that here we are using who to ask for the subject.
We also use who as an object in questions.
Who is she going out with?
Who does she love?
Who are they following?
Whom is possible here, but sounds rather stiff and formal.
With whom is she going out? (We prefer to use prepositions before whom)
Whom does she love?
Whom are they helping?
As a subject of a defining relative clause
Last night I saw the teacher who teaches us technology.
Who is the subject of the relative clause:
I saw the teacher. The teacher teaches us technology.
We cannot use whom, nor can we remove who.
As an object of a defining relative clause
That's the teacher who I saw last night.
Who is the object of the relative clause:
That's the teacher. I saw the teacher last night.
In this case, we can use whom (more formal) instead of who, or leave it out altogether.
That's the teacher who/whom/- I saw last night.
As a subject of a non-defining relative clause
Non-defining relative clauses are more common in a formal style, especially in writing.
I saw Mrs. Potter, who teaches us technology, last night.
We cannot remove who nor can we use whom instead.
As an object of a non-defining relative clause
That's Mrs. Potter, who I saw last night.
Since non-defining clauses are more formal, and we prefer whom in formal styles, we often replace who with whom here.
That's Mrs. Potter, whom I saw last night.
In cases where a preposition is present, we prefer it before whom.
He is very angry with Paris, who he had an argument with last week.
He is very angry with Paris, with whom he had an argument last week.
Valerie, who I told you about just now, is getting married.
Valerie, about whom I told you just now, is getting married.
Here's a group from the Swinging Sixties, Juicy Lucy, performing a Bo Diddley tune, 'Who do you love?'