I cannot figure out whether the sentence "It is the small house that we can replace with a big one. " is a relative clause or an emphatic sentence.
If, setting the terminological inaccuracies aside, I understand the spirit of this question, Alisa means to ask whether the sentence is identifying some house as "the small house that we can replace with a big one" or whether the sentence is clarifying/emphasizing what it is that "we" can replace with a big house.
Her teacher seems to think that it can only be interpreted in the latter way (essentially, as an "it" cleft), in which case the sentence must presuppose that there is, in the context, something that "we" can replace with a big house, the cleft sentence identifying that thing as being the small house.
However, Alisa is right, if I hear her correctly through her terminological inaccuracies, that the sentence can also be interpreted as performing an identification. On this interpretation, "It" refers to something in the world and could be replaced by "This (house)" or "That (house)."
This is the small house that we can replace with a big one.
That is the small house that we can replace with a big one.
It is obvious that those two sentences are not different ways of saying "We can replace the small house with a big one." However, on the other ("it"-cleft) reading of "It is the small house that we can replace with a big one," the sentence is just another way of saying "We can replace the small house with a big one."
By using that other way of saying the sentence, the speaker focuses the direct object in the basic sentence from which the cleft sentence derives. This focusing has an emphatic quality, as evidenced by the fact that, in live speech, the part of an "if"-cleft following "It is" naturally receives stress.
It is the small house that we can replace with a big one.