Why did not delete the ‘it' in the sentence ' Unless the owner can present recent certification that the house is free of termites, have a professional check it.'
I think this sentence is not right unless the 'it' be deleted.
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Because you want the professional to check the house. This is a transitive sense of the verb "to check" and so it needs an object. The object can be a pronoun to avoid repeating "the house".
You could drop the word "it", as the verb "check" can be used both transitively and intransitively. However without "it" the verb feels less clear. It seems uncertain if the professional is checking the house, or if the professional is checking whether the owner can or can't present certification. I don't think there would be any great ambiguity, but including the object as a pronoun helps the expression.