Nothing to do with 'in' I think.
Both are correct, but have subtly different meaning. 'It is in this house that he was born' is a complete statement. It's a long-winded way of saying 'He was born in this house', with added emphasis that he was born in THIS house, not any other.
'It is in this house where he was born' demands continuation - 'It is in this house where he was born that he first saw a rabbit'. Probably surrounding the modifying phrase with commas would be good. 'It is in this house, where he was born, that he first saw a rabbit'
You could also have the neutral statements 'This is the house where he was born'. Or 'This is the house in which he was born'. Or, somewhat antique, 'This is the house wherein he was born'. Or, slightly less ungramatically but colloquially perfectly OK, 'This is the house he was born in'. If your tutors object to this last one, tell them a native English speaker says they're being unnecessarily pedantic!