Can I say
Mr. Smith is a teacher that I really like.
OR Mr. Smith is a teacher whom I really like.
Can I say
I'm not really sure what you mean by "dative" in English, as there isn't really an accusative/dative distinction - in situations where other languages might use a dative, either the accusative is used ("I gave him the book") or a preposition ("I gave the book to him"). However, the following might be helpful in articulating why "who" can be used, and may even sound better, where some insist on "whom" - whereas in other situations "whom" is still preferable:
To make my explanation clearer (at the expense of much precision, for which please forgive me) I'll refer to two "styles" of English - one very formal (in which the who/whom/whom prescribed by the style guides is compulsory), and one much more colloquial (in which who/who/who rules the roost, and "whom" is seldom if ever used). Very loosely speaking these correspond to English as it was both spoken and written in the past, and how it is most often spoken today; since trends in the written form often follow those in the spoken we might see current written English as being in a transition between the two.
Given all that, the sentence:
The man whom I saw yesterday was tall.
entirely follows the rules of the formal style, and is thus acceptable.
The man who I saw yesterday was tall.
entirely follows the rules of the more colloquial style, and is thus acceptable
However, the sentence
*The man to who I gave the ball yesterday was tall.
grates. This seems to be because it follows neither the rules of the formal style (which would have "to whom"), nor the colloquial style (which would instead have "The man I gave the ball to yesterday was tall", or a variant thereof), and is thus unacceptable in either. Similarly,
*The man whom I gave the ball to yesterday was tall.
also falls between both stools.
Note that this is largely handwaving, rather than a rigorous argument, but it's interesting to note that studies have been performed in cultures exhibiting diglossia (i.e. using "high" and "low" variant forms of what by some definitions could be considered one language, in different contexts) where subjects were shown words or sentences combining features of the "low" and "high" variants. It was found that some of the features were only weakly associated with one variant or another, in the sense that (say) using one from the low variant in a sentence otherwise fully "high" would not render it unacceptable; however other features were "strong" in the sense that a sentence containing features strongly associated with "high" and others with "low" would definitely render the sentence unacceptable. It is possible that, on a much smaller scale, a similar phenomenon is going on here (though of course it would be very bold to assert that this is the case without much more rigorous research!)
Whom should be used to refer to the object of a verb or preposition. When in doubt, try this simple trick: If you can replace the word with “he”’ or “’she,” use who. If you can replace it with “him” or “her,” use whom.
Who should be used to refer to the subject of a sentence. Whom should be used to refer to the object of a verb or preposition.