how can I remove very old masking tape and residue from antique wooden box?

Try Goo Gone. Wipe a little bit in an inconspicuous place just to make sure but it's safe with most finishes.

One good use for WD40 too just soak it and let it sit. After a couple days of soaking the tape will generally just about slide off.

Test some lighter fluid in an inconspicuous place. Ronsonol cigarette light fluid, not barbecue lighter fluid.

Soaks through the paper and leaves no residue.

There have been a lot of different masking tapes since 1940. Solvents that won't harm the wood-box finish include waterless hand cleaner, and maybe (odorless) paint thinner, but NOT alcohol (it would take off shellac). Anything brittle can be carefully scraped, a razor scraper might be useful, just control the edge so it does't dig into the wood.

Boiled linseed oil is a safe way to cover any dull spots, and a little french polishing can make an old finish new again.

Couple days!? Clare, I've found that most goo removers do it in a few minutes.

Perhaps you need to be a little less patient with WD-40 next time. I bet it works a lot faster than you think.

FWIW for anyone lurking, many cooking oils and sprays will do the trick as well as goo-gone.

GooGone damagess some finishes, and you can scratch dried tape off with wd40 pretty quickly - but if it is soaked real well the tape slides of - no damage to the shelac or laquer finish.

I likely left it a lot longer than required - but it worked.

how will you keep it closed without the tape

the tape lends it character

I would start with olive oil if the antique is of real value (financial or sentimental).