Does Dual Wield drain HP?
It was clear that Kirito was keeping his dual-wield ability a complete secret. Nobody knew about it - not even Asuna. This was evidenced by the fact that Asuna questioned Kirito about why he isn't using Liz's sword and suggested that he was "hiding something" from her.

So Kirito hesitated because be knew that using his dual wield in front of everyone would blow that secret wide open. But he had no choice. It was do or die.



Getting back to the question, there is no evidence in the anime to suggest that dual wielding drains HP. But you could infer other drawbacks such as tying up both hands thus preventing the use of a shield.