Is it 'Smokey the Bear' or 'Smokey Bear'?

It is 'Smokey Bear'. There is no "the" in his name. That was added by songwriters to make their song sound better. Check the National Forest Service's history of Smokey Bear and you'll see that his correct name has no "the" in it.

The bear has his own US Postal ZIP Code!

Please see the link below: