How can I clean and restore old interior shiplap planking?
Almost all old shiplap was once painted, so it probably isn't raw wood. If so, it has some resistance to moisture.

You can try to clean it using a mild soap cleaner, such as Murphy's or other soaps advertised for cleaning wooden surfaces. This should remove most of the dust and other water soluble soils (and a bit of the oil/grease based as well).

If that works, you may have a surface that is acceptable as is. If not, you can progress to a solvent based cleaner. Mineral spirits is the least aggressive and will remove most grime, but leave most finishes intact. However, this gives off significant fumes, so you need to be able to adequately ventilate while doing this, and for some time (a day) thereafter.

If is still isn't fresh enough, you are now in the realm of heavy duty strippers, and this needs a whole different analysis.

If you get a satisfactory result with one of the above approaches, you can just leave it, or you can try to seal it with a flat or satin based clear coat, such as a water based poly.