Maybe? But probably not.

The physics of the problem are such that it is mathematically possible for a few hundred mirrors to generate enough heat to set wood alight. Modern concentrated solar power generators use the method to generate power today. Frequently this includes melting salt and using the molten salt as for power generation. So conceptually it is possible.

Several attempts to recreate the supposed event have had mixed results. In 2009, 127 MIT students used the method to successfully burn down a rowboat. In 2010 Mythbusters used 500 volunteers, but weren't able get their boat to catch on fire. Even in the MIT experiment it took 10 minutes for the fire to catch. It's hard to imagine that the Romans would just sit there and let Archimedes burn ships down. Supposedly this was done at bow shot range. So you'd imagine the Romans shooting bows to disrupt the mirrors and their holders, and using fire suppression techniques. Unlike Greek fire, mirror started fires would be easily doused.

What casts more doubt on the story is all the surviving ancient sources neglect to mention the mirror story. Livy, Polybius, and Plutarch all wrote about the siege. They all mention "The Claw," which was a crane Archimedes built to catch and flip the Roman ships. They also mention novel new siege engines, catapults and the like. But none of them mention fire or mirrors. Lucian and Galen mention fire, but not mirrors. Our source for the mirror story comes from the 12th century, and although they claim to be quoting an ancient source, that source has been lost to us.

If Archimedes tried to burn the Roman ships, its likely he would have used a more traditional weapon like catapulting pots of burning pitch on to them. The laser beam like weapon that Zonares and Tzetzes describe likely never existed.

The oldest sources don’t say anything about Mirrors
Polybius, writing somewhere in the 100’s BC says:

Archimedes had constructed artillery which could cover a whole variety of ranges, so that while the attacking ships were still at a distance he scored so many hits with his catapults and stone-throwers that he was able to cause them severe damage and harass their approach. Then, as the distance decreased and these weapons began to carry over the enemy's heads, he resorted to smaller and smaller machines, and so demoralized the Romans that their advance was brought to a standstill. In the end Marcellus was reduced in despair to bringing up his ships secretly under cover of darkness. But when they had almost reached the shore, and were therefore too close to be struck by the catapults, Archimedes had devised yet another weapon to repel the marines, who were fighting from the decks. He had had the walls pierced with large numbers of loopholes at the height of a man, which were about a palm's breadth wide at the outer surface of the walls. Behind each of these and inside the walls were stationed archers with rows of so-called 'scorpions', a small catapult which discharged iron darts, and by shooting through these embrasures they put many of the marines out of action. Through these tactics he not only foiled all the enemy's attacks, both those made at long range and any attempt at hand-to-hand fighting, but also caused them heavy losses. - Polybius, Histories

Livy, writing in the 10’s BC wrote:

Archimedes countered by moving into position on the walls pieces of artillery of varying size; at the ships off-shore he hurled stones of enormous weight, assailing those closer in with missiles which, though lighter, could for that reason be discharged more frequently. Then, to enable his own men to discharge their missiles at the enemy without danger to themselves, he made rows of loopholes, ranging from top to bottom of the wall and some eighteen inches wider, through which, themselves unseen, they could shoot at the enemy either with arrows or with smallish catapults. Some of the enemy ships came close in-shore, too close for the artillery to touch them; and these he dealt with by using a swing-beam and grapnel. The method was this: the swing-beam projected over the wall and an iron grapnel was attached to it on a heavy chain; the grapnel was lowered on to a vessel's bows, and the beam was then swung up, the other arm being brought to the ground by the shifting of a leaden weight; the result was to stand the ship, so to speak, on her tail, bows in air. Then the whole contraption was suddenly let go, and the ship, falling smash as it were from the wall into the water - Livy , History of Rome

And Plutarch, writing in second century AD says:

Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships sunk some by the great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. - Plutarch, Parallel Lives

Wasn’t ‘til way later that mirrors & death-rays were added to the story