The Romans never lost it (unless you count 395 CE, when it became part of the eastern empire!). The Romano-Greek inhabitants in antiquity certainly did identify it with the setting of the Trojan War legend; though it's perhaps prudent to point out that at least two other towns in the area also claimed to be Troy. But the site we know was Troy, and which the Romans knew as Ilium, was always the main candidate. It was a fairly important locus of east-west relations, and later even more important as a locus of propaganda for Roman-Greek relations (since the Romans believed they had Trojans in their ancestry).

When Xerxes invaded Greece in 480 BCE he stopped off at Troy on the way there and made offerings to "Ilian Athena" as a propaganda gesture (as though he were coming to avenge Priam!). In 386 BCE all Greek cities in Asia Minor were ceded to the Persian Empire; when Alexander captured Ilion back in 334, he made offerings similar to Xerxes' ones, and gave explicit orders that the site should have a special status, as well as a new temple to Athena (which was completed by Lysimachos several decades later). From 306 Ilion was the capital of a league of cities in the Troad, with ritual games in honour of Athena. This was the highest its prestige ever reached; the Seleucid king Antiochos III again made promises that Ilion would retain its privileges, and he made offerings there following in the footsteps of Xerxes and Alexander. When the Romans walloped Antiochos in 190 BCE, the consul Cornelius Scipio did the same thing again, and the Romans gave the city more territory. There may have been a bad hiccup in 85 BCE: there's a story that a mutinying Roman commander, Fimbria, sacked the city, and boasted that he had done in ten days what it had taken Agamemnon ten years to do; but there's no archaeological evidence to support this. Anyway, Julius Caesar continued the tradition of honouring the city, and even considered making it his capital; and another new temple to Athena was built during the reign of the first Roman emperor, Augustus. They had the strongest possible motivation for this, since they claimed direct descent from Aeneas.

In the imperial period Troy/Ilium reached the biggest size it ever achieved. In late antiquity it declined, but went on to be the seat of a minor Byzantine bishop. It wasn't abandoned until ca. 1306, when the Ottomans invaded. After that, the site was indeed forgotten.

It was first re-identified with classical Ilium in the modern era by Edward Daniel Clarke in 1801; Charles Maclaren identified it with "Homer's Troy" in 1820. Some brief excavation was done by a British engineer, John Brunton, when he was stationed nearby during the Crimean War (1853-56). The first people with archaeological experience to visit the site were Frank Calvert and Johann Georg von Hahn in 1863-65, but they didn't get down to any Bronze Age material. Then Schliemann did major excavation work in 1871-72, 1878-79, and 1882-83; and, working under Schliemann, Wilhelm Dörpfeld sorted out the stratigraphy and did what he could to recover the data from Schliemann's haphazard and destructive processes. Though Schliemann liked to pretend that there was an entrenched orthodoxy against him, there was never any real doubt that it was Roman Ilium/Greek Ilion, and the city that the Romans themselves identified as Troy.