Wicker tower shields (called gerrha in Greek) were indeed used by the Immortals and other Eastern infantry. A rare nearly-intact example was excavated at Dura Europos in the Syrian desert; it consisted of sticks woven through a leather facing. Smaller wicker shields were used even by Greek light troops, as they were extremely lightweight and relatively easy to make.

I'm not aware of any penetration testing on this type of shield. However, M.M. Khorasami (Arms and Armour from Iran (2006)) claims that these shields would have been remarkably strong and resilient. While some soldiers depicted on Persian reliefs were armed with round shields made of wood, the wicker shield never seems to have gone out of style in Antiquity, suggesting that it remained effective enough.

The primary use of the Immortals' tower shields was to form a shield wall. Only the first rank of their formation carried these shields, which they set down in a continuous line once they got within arrow range of the enemy. From behind this shield wall, the rest of the formation would then start raining down arrows on their opponents. The shields, then, served both as a protection from enemy missiles and as a barrier to any shock troops (and especially cavalry) hoping to dislodge the archers. The Persians did not skirmish, like Greek missile troops; they did not give ground when attacked, but dropped their bows and fought as heavy infantry. Their shield wall served to bolster their defence in close combat.

Both at Plataia and at Mykale, Greek heavy infantry charged the Persian lines as a desperate response to their lethal volleys, but they had to tackle the shield wall before they could come to grips with the Persians themselves. This seems to have broken their charge on both occasions. However, once the shield wall was down, the Persian defence lost much of its effectiveness, as the Greek heavies all had large shields to protect them in hand-to-hand combat.