The Trojan War, if it ever happened, was a millennium before Alexander's time. Would folk memory of Achilles' resting place have survived so long? Do archaeologists have any clues as to where the tomb is?
edit: Just found this analysis of the question while googling some of the stuff posted in the answers. It turns out the question is way more complex than I thought, as it's impossible to isolate it from attempts to ground the Trojan War in fact rather than legend, but there are several contending locations for the tomb of Achilles. None of the ones we know about seem like particularly compelling candidates for the tomb of a Mycenaean warrior of the late bronze age to me, but oh well. It was an interesting read regardless.
Why was that tomb lost, and did it ever really hold Achilles' remains?
The question assumes the tomb's existence. You should rephrase it as "did the tomb actually exist?" You could, and should, go further by asking "was there an Achilles and what could we know about him?"
Plutarch lived at the turn of the age, about 300 years after Alexander's death. What Plutarch knew of Alexander he knew from earlier sources. If you've read his Alexander, you'l recall where he says "It is not histories I am writing, but lives." His Life of Alexander is less a history than a collection of stories portraying the important virtues (and vices, Plutarch was a moralist) of the character in question, in this case Alexander. Though Plutarch wasn't by any means a bad historian, in this case he wasn't much concerned with whether the earlier reports of Alexander's visit were historically accurate but rather what the event revealed about Alexander's character. Plutarch is writing a narrative life of Alexander some three centuries later, in which he cites a probably true story about Alexander sacrificing at what people of the time believed to be Achilles' tomb, this some 800 years after the alleged Achilles.
Was there an Achilles? Certainly not, not if one means the demigod offspring of the nymph Thetis and Peleus king of the Myrmidons. But there may have been a hero of the Trojan War who served to inspire the myth.
So now we ask "was there a Trojan War and what could we know about it?" Thucydides, (ca 5th century BCE) believed there was but also that Homer - who wrote his tales not a few centuries after the alleged events - and the other accounts were more poetry than history. At about the same time, Euripides recast many of the old myths freely. In the 19th century the west mostly agreed the war was pure myth but then Schliemann touted his Hisarlik excavations as the historical Troy. Nowadays a lot of scholars agree that there is some core of historicity in a "Greek" expedition against Troy. But no one would venture to say what, if any, of the mythical events are in fact historical.
Finally, the traditional site of Achilles tomb is at Beşik Tepe near the ancient podunk of Achilleion ("the place of Achilles"). The archaeological record of the place begins in the 6th century BCE. While that doesn't prove the place wasn't already there it is a strong indication, and further jibes with Herodotus' (5th century BCE) dating of the place (in his account of the Lelantine war).
Which brings us to the question of folk memory vis a vis Achilles resting place. I suppose it's possibly but how likely is it that a thousand (or even four hundred) year long game of telephone produces at the end anything that can be called "historical?" To be sure, we can say with varying degrees of certainty what some people at some particular time might have believed to be true (historical) but it is difficult if not impossible to recover any actual original history.
Have I answered your question?
Our main source for information on the Trojan War comes from the epic poems of the Iliad and the Odyssey, which seem to have been formed in the eighth century BCE. The poems purport to tell the story of a Bronze Age era war between Greece and Troy, and there is some evidence for violent destruction at Troy near the end of the Bronze Age, but we can't be sure that any Greeks had a hand in it.
The Iliad is not history, it's poetry. While it can tell us some things about the Mycenaean period in Greece, it is probably more helpful in telling us about eighth century society. It is in the eighth century when the heroes of the Trojan War begin to be honoured. At Sparta, for example, a sanctuary to Hellen and Menelaus was established, and at Mycenae there was a cult to Agamemnon. In this period as well, people in Greece begin to perform rituals at Mycenaean tombs, and even reopen and reuse them in order to connect themselves to the Heroic past.
Troy was only lost during the Medieval period, and several prominent ancient figures visited it, and made sacrifice to the dead heroes, including Augustus and Alexander. Any tomb that Alexander saw when he came to Troy was almost certainly not the tomb of Achilles. Names of heroes that have been attached to real archaeological material are usually anachronistic, they were named that after the popularity of the Homeric epics, not before. This is the case at Mycenae, when Schliemann excavated there he claimed the large Tholos tombs were the tombs of Atreus and Agamemnon's wife Clytemnestra, this is what the Classical Greeks believed as well, but in reality these tombs date to about 400 years prior to the supposed events of the Trojan War.
Source: Antonaccio, C.M., 1995. An Archaeology of Ancestors: Tomb Cult and Hero Cult in Early Greece. Rowman & Littlefield Publishers, Inc.