The short answer is “Yes, if it’s the right oil, and you let it dry first.”
Here’s a detailed answer for those interested.
There are two general classes of oils. Drying Oils, and Non Drying Oils.
A Drying Oil is one that will harden to form a solid film after exposure to the air over time. The oil hardens through a chemical reaction called cross-linking, or polymerization by intaking oxygen. So in actuality they “cure” rather than dry. Just like a polyurethane.
Some examples of drying oils include tung oil, linseed oil, and walnut oil.
Note: Danish oil is not an oil per se. Rather, it is a combination of two things; a drying oil, and a varnish. There is no fixed formula for a Danish oil. Each manufacturer creates their own blend. And so can you. The guideline is that it’s typically 1/3 varnish, and 2/3 oil.
This combination is a drying/curing combination that will leave a hard film finish.
2nd Note: Varnish is not a specific product. (e.g. polyurethane, lacquer etc.) Varnish is the term used to talk about the appearance of a finished piece. For example; “that table has a nice varnish,” or “that table has a nice glossy hard finish.”
To be considered a varnish, the products used must be clear or translucent to let the wood grain show through. Paint is not a varnish.
A varnish (or varnished look) can be achieved with products or combinations of products that include polyurethanes, lacquers, epoxies, etc.
In short; you CAN apply a lacquer or a polyurethane over a Danish oil. Just as you can over any Drying Oil. The key to success is to make sure the oil has fully dried first. This should make sense, because once dried, the oil has cured to a hard solid film surface, and stuff will now stick to it.
The other general class of oils are Non Drying Oils. These are oils which will not polymerize & harden with exposure to oxygen. And they will stay “wet” or “oily” forever.
The Real Difference Between Drying & Non Drying:
Chemically speaking, (among other things), oils have long chains of carbon-carbon bonds. The bonds between these carbons are either double bonds, or single bonds. If all of the bonds are single, we say the compound is fully saturated. If some are double, we say it is unsaturated.
We can measure just how many double bonds there are. If there is a lot, we can say it is very unsaturated.
In organic chemistry, we can assign a measure to the compound designating just how many single bonds there are vs double bonds. And we can calculate the degree of unsaturation. For oils we typically do this by reacting iodine with it. The more iodine that reacts means the more unsaturated it was.
A compound with a high iodine number has a lot of unsaturation, and is therefore a drying oil.
A compound with a low iodine number has a low unsaturation and is therefore a non drying oil.
The guideline is this:
Oils with an iodine number of less than 115 are considered non drying.
And oils with an iodine number greater than that are considered drying.
Tung oil - 163-173 - Drying
Walnut oil - 120-155 - Drying
Linseed oil - 136-178 - Drying
Mineral oil - 0 - Non Drying
Coconut oil - 7-10 - Non Drying
Palm oil - 16-19 - Non Drying
A table of some values:
In summation, just make sure the oil you applied to your piece was a drying oil. Then make sure that it has had the time to fully dry/cure (days to weeks depending on heat & humidity). After that, feel free to add a top coat of lacquer or polyurethane to it.
In contrast. If you used a non drying oil, such as mineral oil, your piece will not take a topcoat of lacquer or polyurethane in the future, no matter how long you wait.
Finally, if you don’t know if the oil you used was a drying oil or not, just look up the iodine value on a table. Anything over 115 is a drying oil.