Can I apply tung oil on a piece of wood which has been stained with water-based dye?

In my experience with water based dyes (Like Aniline or alcohol based dyes, etc.) the water “carries” the dye into the pores of the wood.  After it dries, you need to “lock in” the dye with a sealing agent.  In this case, Tung oil.  There are other finishes that will work too, but you have to consider the use of the wood (Marine-commercial-decorative??) and the appearance you want to achieve.
Tung oil finishes usually take a day or two to dry, unless you rub it in several thin coats.  It is also good for oily woods that need to be restored, as the oil soaks into the wood, renewing it’s luster.  Unless you are familiar with it, make sure you test it on a piece of scrap.  make sure you blow off the piece with compressed air or use a tack rag to clean the open grain prior to finishing.  Tung oil is not a thick “top-coat” finish like polyurethane, it’s usually rubbed into the pores of the wood to seal it.
Here is an Article on using Tung oil as a wood finish: Tung Oil – Woodwork Details

Tung oil is usually a wiped on finish; it’s a good idea to wear gloves and have good rags for wiping and cleanup!  Remember the oily rags are combustible, so hang them or dispose of them carefully.

And here are some things you may want to have on hand:
Now you know how I would do it!  Best of luck when using Tung oil!
I hope you found my point of view enlightening and informative.