Well, no.

Temperature is the thermodynamic quantity that is shared by systems in equilibrium. "Cup of tea + information about all the molecules in the cup of tea" is in thermodynamic equilibrium with "Ice cube + kinetic energy (e.g. electricity)", in that you can arrange a system where the two are in contact but do not exchange any net energy.

Note that it is NOT in thermodynamic equilibrium with anything hotter than an ice cube, as Eliezer described in spxtr's linked article: http://lesswrong.com/lw/o5/the_second_law_of_thermodynamics_and_engines_of/

Basically, if you, say, try to use the information about the water and a Demon to put the system in thermal equilibrium with some warm water and electricity, you'll either be prevented by conservation of energy or you'll wind up not using all the information at your disposal. And if you don't use the information it's as if you didn't have it.

The salient point is that the system is not in thermal equilibrium with anything 'warmer' than "Ice cube + free energy."

If you know everything about the cup of tea, it really is at absolute zero, in the realest sense you could imagine.