Depends, doesn't it?
Who shall I shoot? Better them than me!
Who shall do the washing up? Better they than I!
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Well, if you buy the argument that "He is better than they," is short for "He is better than they are," than use "they". Else, you should be okay saying "them", but you're risking traditionalists jumping all over you. I'm surprised, though, that your other teacher calls "they" incorrect.
It's true that "I" is the subjective case, but there are many cases where pronouns occur in a position that is neither subject nor object. Whether your interpretation of the original sentence ("He is better than they.") is one of those, depends on whether you buy into the argument that it's short for "He is better than they are." If you don't feel the implied "are", the difference is not as clear.
It's more obvious with Rich's example:
This is I.
The pronoun, here, is neither a subject nor an object, but a subject complement. In related lanugauges, such as German, subject complements take the same case as a subject, so one could argue by analogy that "This is I," is correct. But English isn't German, and hardly anyone would say "This is I."
However much you like by-the-book correctness, a book that tells most of the native speaker's intuition that they are wrong is a silly book.
My Oxford Dictionary of English has this to say:
Oxford Dictionary of English said:
Where a personal pronoun is used alone without the context of a verb or a preposition, however, the traditional analysis starts to break down. Traditionalists sometimes argue, for example, that she's younger than me and I've not been here as long as her are incorrect and that the correct forms are she's younger than I and I've not been here as long as she. This is based on the assumption that than and as are conjunctions and so the personal pronoun is still subjective even though there is no verb (in full form it would be she's younger than I am). Yet for most native speakers the supposed 'correct' form does not sound natural at all and is almost never used in speech. It would perhaps be more accurate to say that, in modern English, those personal pronouns listed above as being objective are used neutrally - i.e. they are used in all cases where the pronoun is not explicitly subjective. From this it follows that, despite the objections of prescriptive grammarians (whose arguments are based on Latin rather than English), it is standard accepted English to use any of the following: Who is it? It is me!; she's taller than him; I didn't do as well as her.
Basically, this means you can choose, but you'd better be careful before calling the alternate usage wrong. And if you know the person you're writing to likes traditional grammar, do them a favour and use the subjective case. And when you're speaking to normal people do yourself a favour and speak the way your tongue tells you to. They won't hate you for it. (The same should go for writing; although there are probably overzealous editors out there.)
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Edit: Heh, I just noticed I must be a traditionalist. "Who shall do the washing up? Better they than I." is a typical traditionalist argument. According to the Oxford quote above "Better them than me," should be acceptable. (But I do see the subjective case implied in the reply as an answer to the question. Oh, well...)
