I just saw a MythBusters episode where built part of a trireme and attempted to burn it using
(a) archimedes death ray
(b) burning arrows.

In both cases their result was that it was very difficult to burn a trireme despite being made of flammable material- wood, tar etc

Just Ram it!!

Very interesting (pitch rather than tar, I think ?)

The Mythbusters were specifically trying to duplicate the "Archimedes Death Ray", which they have done at least one other time.
Pretty much proved it couldn't work even with modern technology. (real mirrors instead of polished bronze)

They resorted to the fire arrows mostly for fun....

From other historical sources, the likely burning agent used in naval battles of the time was something similar to Greek Fire, which could be simply flung at the enemy in pots or pumped from ship-mounted projectors.
Historical models of such pumps have been built and function very well.... Early flame-throwers.

I do wonder about the flammability of triremes in battle-- they went into combat without masts and sails, so basically it's timbers, caulk, pitch (mixed with wax) on the bottom of the hull, leather for the rowers' seats and oar thongs, etc.

I was not clear.

1. I don't think triremes burnt easily: I can't see what would burn (no sails).

2. I tried to think of arguments in favour of Bill Skinner: caulk ? Resinous timbers ?

3. I can't think of any examples of triremes being set on fire during battle. Here's an extract from Diodoros 19, the big seabattle at Salamis (306 BCE):
51 1 The princes, since they were about to fight for their lives and their all, were in much anxiety. When Demetrius was about three stades32 distant from the enemy, he raised the battle signal that had been agreed upon, a gilded shield, and this sign was made known to all by being repeated in relays. 2 Since Ptolemy also gave a similar signal, the distance between the fleets was rapidly reduced. When the trumpets gave the signal for battle and both forces raised the battle cry, all the ships rushed to the encounter in a terrifying manner; using their bows and their ballistae at first, then their javelins in a shower, the men wounded those who were within range; then when the ships had come close together and the encounter was about to take place with violence, the soldiers on the decks crouched down and the oarsmen, spurred on by the signalmen, bent more desperately to their oars. 3 As the ships drove together with force and violence, in some cases they swept off each other's oars so that the ships became useless for flight or pursuit, and the men who were on board, though eager for a fight, were prevented from joining in the battle; but where the ships had met prow to p285prow with their rams, they drew back for another charge, and the soldiers on board shot at each other with effect since the mark was close at hand for each party. Some of the men, when their captains had delivered a broadside blow and the rams had become firmly fixed, leaped aboard the ships of the enemy, receiving and giving severe wounds; 4 for certain of them, after grasping the rail of a ship that was drawing near, missed their footing, fell into the sea, and at once were killed with spears by those who stood above them; and others, making good their intent, slew some of the enemy and, forcing others along the narrow deck, drove them into the sea. As a whole the fighting was varied and full of surprises: many times those who were weaker got the upper hand because of the height of their ships, and those who were stronger were foiled by inferiority of position and by the irregularity with which things happen in fighting of this kind. 5 For in contests on land, valour is made clearly evident, since it is able to gain the upper hand when nothing external and fortuitous interferes; but in naval battles there are many causes of various kinds that, contrary to reason, defeat those who would properly gain the victory through prowess.

52 1 Demetrius fought most brilliantly of all, having taken his stand on the stern of his seven. A crowd of men rushed upon him, but by hurling his javelins at some of them and by striking others at close range with his spear, he slew them; and although many missiles of all sorts were aimed at him, he avoided some that he saw in time and received others p287upon his defensive armour. 2 Of the three men who protected him with shields, one fell struck by a lance and the other two were severely wounded. Finally Demetrius drove back the forces confronting him, created a rout in the right wing, and forthwith forced even the ships next to the wing to flee. 3 Ptolemy who had with himself the heaviest of his ships and the strongest men, easily routed those stationed opposite him, sinking some of the ships and capturing others with their crews. Turning back from that victorious action, he expected easily to subdue the others also; but when he saw that the right wing of his forces had been shattered and all those next to that wing driven into flight, and further, that Demetrius was pressing on with full force, he sailed back to Citium.

Even with engines on board, no fire.

4. Greek fire is a later invention than the great age of oared fighting fleets.