From Wikipedia article on the Temple of Trajan in Rome:
The Temple of Trajan was a Roman temple dedicated to the emperor Trajan and his wife Plotina after his deification by the Roman Senate. It was built in the Forum of Trajan (Rome), by Trajan's adoptive son and successor Hadrian, between 125 and 138. It was destroyed in the Middle Ages.
The same article states that the exact place of it is unknown.
Is it known how and when exactly was destroyed?
Google searching this can be tricky, as most results are from a Trajan Temple in Pergamon (destroyed by an earthquake), and from another Trajan Temple in Antioch (burnt down by Emperor Jovian).