"Can I can paint on an oil painting that I started two years ago but never finished? Someone told me that I should simply varnish what I'd done, but the painting isn't finished." -- Sharona
"Is it possible to paint in oils over an oil painting? I have unearthed some old canvases that are a bit worse for wear and rather dirty. Would I have to coat them with gesso first?" -- Serefosa
You can paint on an old oil painting like it was a new one, just make sure there is no grease or dust on it, though personally, I wouldn't do anything but a study over a really, really old canvas. (Ask yourself, is it worth the slight risk an old painting showing through, or the new painting cracking because the old painting pulled in all the oil, for the price of a canvas?) If the colors on the old painting are very intense, consider painting a layer or two of fat titanium white (not gesso) on top, letting it dry completely between coats.
Dust can be dealt with by wiping the painting with a damp cloth and leaving it to dry. If there's any grease on the painting, I'd wash it lightly with a solution of rubbing alcohol and let it dry overnight. You can then paint on it as usual to finish it. Just don't overwash it. If you painted it okay before there shall not be any problem, but if you notice colors on your rag just be cautious when cleaning.
You can "wake up" an old oil painting before you continue painting on it by applying on a thin coat of oil medium and leaving it to stand for at least a day (somewhere where it's not going to gather dust!). The technical term for this is oiling out. Remember also, that the new oil paint you're going to apply has oil in it which will also 'feed' the old paint.
Some Old Masters, when glazing, used did a thin "wake up" layer between dried coats.