Could I refinish it by simply applying a drying oil (tung or linseed) over the mineral oil?
The standard answer to this on woodworking forums would be no, but in reality it might work acceptably. However, it's likely to compromise the ability of the drying oil to 'dry' properly, and this effect will be permanent, so it's safer to work to remove as much of the mineral oil as possible before applying any other finish.
Since mineral oil, like all oils, is easily soluble in mineral spirits (UK: white spirit) it can be used to rinse a lot of the oil from the surface wood fibres.
You'll want to use generous quantities of spirits and plenty of fresh cotton rags or paper towels to soak up liquid, perhaps doing three rinses and wipe-downs and then leaving the table to dry.
The next day check that the surface looks fairly uniformly de-greased and locally treat any areas that still look wetted with oil.
Safety note: normally the fumes from spirits are not a big deal but we're not usually using them in quite this quantity so if possible do this job outside. If that's not feasible work in the most well-ventilated place possible; a garage with the door wide open would be good if available. Even after the rags/paper are bagged up and thrown away the table itself will give off a lot of vapour as it's drying out so you wouldn't want to leave it in any room you're using for another purposes.
Once the cleaning is complete you can apply the drying oil of your choice. It won't hurt to thin an amount of the oil with spirits for the first couple of coats; a 1:1 dilution works well. My recommendation for the oil would be BLO over tung oil as it makes the wood look better faster, is easier to get and costs less.
Incidentally you can also use varnish, or an oil/varnish blend, to provide better durability and water-resistance.
Incidental point on wood removal:
Or would I need to completely sand off the layer of wood that soaked up the mineral oil before refinishing it?
For the average home woodworker sanding should really be the method of last resort to remove a layer of wood from a large flat area, it's generally only a good method where there is access to a wide-belt sander. When working at home if you're thinking of taking off a layer of wood your first thoughts should be plane or scraper, both of which are faster and more efficient than sanding and produce much less fine airborne dust.