If it looks good now, the finish matches and the colors are good, I
would leave it. You answer is an easy one from the stand point of what
Where the dog chewed off the wood, it removed the old finish on the
wood and exposed a new, pourous surface. The tung went right in in the
wood with no problem.
But where the wood still has some finish on it, it makes the tung oil
sit on top of the old finish without penetration, or maybe just a small
amount of penetration. In your case, you are now waiting for the new
finish (over the old) to catalyse by air exposure alone.
I would let it sit in the house where it is warm and presumable if it
is cold outside, your heating system will be busy removing the moisture
from the air. Again, if the finish matches, that is the most important
thing. You may be waiting a few days for this finish to really harden
If you get impatient and try to remove it, you will probably
dissolve/damage the old finish with and wind up refinishing the whole