The traditional oil finish for furniture is turps, boiled linseed (and sometimes a drying agent to help it cure a bit).
It's called a varnish, because it does form a very thin film finish, but it is primarily just soaked into the wood. It's an excellent finish for any furniture, and it fairly durable, and is resistant to heat and moisture damage, unlike most other traditional finishes. It's very easily applied (wipe on with a rag, then after 15 minutes or so wipe off the excess with a clean rag, and repeat a couple times a day). The downside is that it takes many many coats to build up a nice finish. Like 10-20. And like all oil finishes, it will darken the wood with time and exposure to sunlight.
If all you want to do is clean the wood, you could use a furniture restoration cleaner like vulpinex (which will not harm the original finish on the furniture, but will remove dirt and oils on the surface - this is the stuff used by museums to clean and brighten their 200 year-old aquisitions without changing the historical finish on the piece). Or you can wipe it down with some turpentine on a rag, which will remove waxes and oils and some dirt, but may also remove some finishes.
Or, you can just clean the furniture with a slightly damp rag (damp with water). That's the most basic and least invasive cleaning for an antique. Don't leave any water standing on the surface, and never have the wag so damp that it drips.