During the American Revolution- No. The Balaclava- a face covering- was adopted by the Army during the Crimean War. However, this was in the middle of the nineteenth century, over eighty years after the 1775 events of the show.

The British Marines (note- they were not Royal Marines until 1802) wore uniforms which were essentially similar to the Army. Variations existed in the facings, lace, buttons, belt plates and other miscellaneous accoutrements, as they did in any of the other regiments. Notably, Marines' lapels buttoned with eight buttons on each lapel. Lace was of an open bastion pattern with a blue stripe and red worm on a white field. Facings (displayed on cuffs, collar and lapels) were white. Within the Marine divisions there were further distinctions. The Chatham Division, for instance, wore a black plume. I know of no reference to balaclava or similar garments being used by the Marines. I'm going off the documentation in the appendix of the first volume of Cyril Fields' Britain's Sea Soldiers- a work which was compiled prior to the First World War.

The Marines served in detatchments aboard every Naval vessel with ten or more guns. As such, they went wherever Naval vessels went and would have been exposed to hugely varied climatic conditions in the process. Certainly more variation than Army regiments would experience in the course of a single campaign. Cold weather gear would have been obtained for the Marines from the Ships' slops, as was their working clothing. Whether this included some form of face covering- perhaps simply a scarf worn about the face- I again cannot say.

If you want my take on this particular historical 'liberty', amongst the innumerable others in this series- the masks turn the British soldiers into so many faceless storm troopers. Even in the historically awful movie The Patriot, which treated the British army in the Revolution as so many mindless automatons led by aristocratic sadists, left the soldiers' faces uncovered. Apparently leaving even a veneer of humanity on the British Army is too much for the History Channel- the British Army must be archetypical movie bad guys, and like movie bad guys from bandana wearing train robber in the Wild West to the Orcs in the Lord of the Rings, they are rendered in some way faceless.

The best work- though now over forty years old- on British uniforms during the period of the American Revolution is Hew Strachan's British Military Uniforms 1768-1796. More recently, Matthew H. Spring's With Zeal and Bayonets Only is a superb study of how the British Army really fought in America during the years 1775 to 1783.