My first instinct was no way, remembering that the branches and leaves contain a high amount of thujone, which is a neurotoxin and not without risks, especially if used over a long time or while pregnant. This is the same stuff that caused absinthe to be discredited for decades.
But Thuja oils typically contain 40% α-thujon, sage (salvia officinalis) up to 60%. Thuja's pungent smell would prevent ingesting too much.
So my conclusion would be:
do not use for a long time or when pregnant (most midwives caution against using sage, too, btw.)
use sparingly for health reasons, a limited amount is considered safe by health agencies world wide, see information on absinthe for example.
use very sparingly for culinary reasons, because the taste is quite "pronounced". I had to take a herbal medicine with thuja decades ago and still shudder to think of it. (It did work, though...)
When thinking of potential culinary uses, think of uses for sage or juniper - you will want to use it as a spice in foods that can "handle" the woody, bitter tinge. Dark red meat or even game perhaps? Complement with a generous amount of pepper and perhaps a good red wine.
Another approach that might be worth a try is sweetening it up - like "fir syrup" (is this known in American cuisine?).