I have an old table which I would like to disassemble and restore. The side and end rails of the table are joined to the legs using a mortise and tenon, which is then skew nailed from the inside of the rails, through the tenon and into the leg. The nail doesn't protrude at either the point or the head end. I'd like to avoid drilling around the nail heads if possible, although I fear this might be the only option.
No option, you're going to have to dig into the wood a bit. You need to be able to get a grip on the nail heads to begin to pull them out and if they're sunk below the surface you can only do this by removing some wood.
With a bit of luck though you may only need to drill two small holes on either side of each nail, just enough to allow the tips of pliers to get a grip. Needle-nosed pliers are ideal if you have them.
Once you've gotten a grip on the nail pull straight up. The grip of wood on a nail can be very tight so this can be difficult at the best of times, but in the Comments you say the table has been in the weather so be prepared for the possibility that this will be much more difficult than you'd like (sorry!) as the nails are likely to be rusted, possibly along their whole length, and rust can virtually glue a nail to wood.
Assuming you can pull the nail up enough that the head projects, obviously continue pulling straight up if the nail is being cooperative.
If it's hard work and the pliers keeps on slipping free (at worst taking the nail head with it) it's wise to switch to using a nail puller, forked prybar (cat's paw) or the trusty claw hammer. These all require you to lever to one side of the nail so to protect the wood from being dented you need to insert a thin bit of wood or plywood. The blade of a putty knife also works well.
Pliers only — if by chance the pliers is the only tool available you can greatly ease the pulling of nails by also taking advantage of leverage. I couldn't find a photo showing this method so I'll have to try to describe it as best I can. Firstly you switch grip, so that the pliers is holding the nail from the side, just under the head ideally. Insert a thin piece of wood or dowel (the end of a paintbrush handle will work well) underneath the hinge area of the pliers to provide a fulcrum, now lowering the grips will rotate the nose of the pliers upwards, pulling the nail with it.
This is amazingly effective, requiring little physical effort (go leverage!) and I've successfully removed rink-shank nails this way when I couldn't get a grip on them with the claw hammer I was using. It can be slow, as you need to lever, let go and grip again, lever, rinse and repeat, but it gets the job done.
The drilling and maybe some of the levering is going to mark up the wood around the site of each nail and it'll look a bit chewed up, but as this is on the inside faces of the apron the damage won't normally be seen. If you're like most woodworkers though you will want to tidy it up. Obviously you can just slap in some filler and call 'er done but a better way is plug with wood. Often this is done with a piece of dowelling but because this is a mortise cheek I think it's probably best to drill a shallow hole (e.g. using a Forstner) and glue in a long-grain plug of similar, or if you prefer contrasting, wood.
In either case this is purely an aesthetic improvement, filled/plugged or not I don't think it will materially affect the stability of the joints once the table is put back together again.
P.S. If after cleaning off the joint faces you find that the fit of the tenons in the mortises is loose you need to carefully choose your new adhesive. None of the usual furniture glues, including all white and yellow types of PVA, hide glue and the common foaming type of polyurethane glue are suitable for a very sloppy joint. I think your best bet would be to use epoxy, which has excellent gap-filling properties.
If you find the straight epoxy is too liquid blend it with wood flour or fine sanding dust to thicken it up (see, What have peanut butter, ketchup, syrup and mayonnaise got to do with epoxy resin?).