Article Three, Section Three of the Constitution defines treason as follows:
Treason against the United States, shall consist only in levying War against them, or in adhering to their Enemies, giving them Aid and Comfort. No Person shall be convicted of Treason unless on the Testimony of two Witnesses to the same overt Act, or on Confession in open Court. The Congress shall have Power to declare the Punishment of Treason, but no Attainder of Treason shall work Corruption of Blood, or Forfeiture except during the Life of the Person attainted.
This sets up two avenues for treason prosecutions. One is the “aid and comfort” path, wherein somebody aiding a country waging war on the U.S. can be charged, and the other is the “levying war” path, wherein one is charged for actively waging war against the United States, or an individual state.
It seems obvious that Snowden’s actions don’t qualify as levying war against the U.S. as all the levying war cases require an assemblage of men and force.
But that still leaves open the “aid and comfort” provision. Even that, however, has its limits. For example, you have to be aiding and abetting a country or entity with whom the U.S. is actively at war.
That’s why Julius and Ethel Rosenberg, when charged with passing state secrets relating to the design of the atomic bomb to the Soviet Union, couldn’t be charged with treason. They couldn’t be charged for treason, because the Soviet Union wasn’t an enemy for the purposes of treason law, because we weren’t at war with them.
Same goes for Aldrich Ames, a former CIA officer convicted of selling secrets to the Soviets, whose actions resulted in the execution of at least 10 Soviet officials who had been working for the U.S. government, as well as Robert Hanssen, an FBI agent whose spying for the Soviets also led to at least three U.S. assets dying.
So the government would have to demonstrate that Snowden was actively trying to provide aid and comfort to a specific entity, such as al-Qaeda, with which the U.S. is at war. What’s more, all treason cases require two witnesses to the “overt act” in question. So the federal government would also need two witnesses who observed Snowden leaking the information.
I don't believe Congress has issued a declaration of war against al-Qaeda. Furthermore, al-Qaeda doesn't represent a nation and is not a nation itself. So I don't think the US could formally be at war with them in the legal sense.