The tree, which stood for more than 200 years in the Northumberland National Park, was found fallen in September 2023 following storms.
Daniel Graham, 38, and Adam Carruthers, 31, have also been charged with causing criminal damage to Hadrian's Wall, the Crown Prosecution Service (CPS) said.
They will appear before magistrates in Newcastle on May 15, it added.
When it was found, the crown of the tree lay partly on the ancient Roman fortification, which stretches 73 miles (118 kilometres) from coast to coast.
The landmark sycamore, located in a dramatic dip in the landscape, became internationally famous when it was used for a scene in the 1991 blockbuster film "Robin Hood: Prince of Thieves", starring Kevin Costner.
It won the Woodland Trust's Tree of the Year in 2016 and was a key attraction that has been photographed by millions of visitors over the years.
Efforts are now under way to see if the tree can be regrown from its stump or saplings from its seeds.
Hadrian's Wall was begun in 122 AD during the reign of emperor Hadrian, and marked the boundary between Roman Britannia and unconquered Caledonia to the north.
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