Tanya says that the product of a multiple of ten and a multiple of ten will always be zero. is she correct

No she is wrong because product of a multiple of ten and a multiple of ten will not always be zero

It depends on what will always be zero. If the last digit is 0 with no value in front of it, then she's considered correct. If the last digit is 0 but it has a value (example: 20) in front of it, then she is considered incorrect.