Almost certainly not. The Constitutional definition of treason in the US is such that it is pretty much never charged, outside of TV shows and movies.
The charges usually discussed in the context of the ongoing US criminal investigation of Assange are violations of the Espionage Act, violations of the CFAA, theft/conversion of US government property, and conspiracy.
OTOH, the government that has actually requested his extradition is Sweden, and not for anything related to national security matters.
As others have pointed out, very unlikely he could be done for treason given he isn't American. But let's not let facts get in the way of some good down voting.
While one might criticize Pres. Obama for his lack of success in closing the detention facility at Guantanamo Bay, I don't think there is any indication that the administration is actually adding detainees there (not that Assange would be the type of person who would have been detained there even when new detainees were being added.)
Also, why would the US not have an extradition request for him open with the UK? The UK would be much more likely to do as the USA pleases than Sweden.
There is a publicly disclosed criminal investigation, and several years ago there were reports based on a non-detailed reference in an email in the Stratfor leak (which didn't come from anyone who would have firsthand knowledge) that there was a sealed indictment produced from that investigation, which the Justice Department has flatly denied.
So, apparently not.
> Also, why would the US not have an extradition request for him open with the UK?
If there were charges, there probably would be an extradition request.
I'm not sure why you think this would be a problem; the regular criminal justice system has processes for handling classified evidence and is the regular venue for people accused of espionage.
> I wouldn't be surprised if they tried to treat Assange as an enemy combatant (i.e. no due process)
The cases challenging detentions under the Bush Administration pretty clearly established that "unlawful enemy combatant" designation comes with its own due process protections (lawful enemy combatants are Prisoners of War, who have rather extensive legal protections.)
No charges against him levied by the US. No extradition warrant out for him by the US.
"Questioning by police" is an enormous understatement. If Assange leaves the embassy, he will be arrested, extradited to the US, and likely convicted of treason. His fate would at best look something like Chelsea Manning's, and at worst, execution.