Hi David, cool idea, weaving from plane shavings. Just a few thoughts about finishing. Plain oils like BLO, oil based varnishes, and oil-varnish blends change colors of wood and can also yellow with age. But water based finishes are often recommended for light wood since they do not change the color much. I would experiment before putting them on a project. I remember Paul using a water based finish somewhere, but I have not had great luck with it. I probably need to experiment more. When I put it on a project it came out sort of glumpy or something. I took it off and tried something else.
A second option might be one of the ultra blond shellacs. You can get flakes from someplace like shellac.net, mix them up with good quality alcohol, and test them out. Even if that turns out to be not quite right, there is no waste since you can use them elsewhere. (Unmixed flakes, in an airtight container, can last a long time, though once mixed, they need to be used in a year or so or they begin to deteriorate.) They add very little color compared, say, to the canned kind of shellac.
Another option for something that is not going to get much wear is just wax. You said this is for a lamp, but I am not clear what part of the lamp your picture shows. I have never just used wax on anything, but I know turners sometimes do.
One other thought. It just occurred to me that your woven plane shavings might be rather flexible. I wonder how flexible they are even though they are in a frame. Most finishes like shellac or varnish are rather brittle. Might those finishes just crack if put on something that flexes even a bit, like a woven mat? If so, that points toward simple wax or some sort of oil (or oil/varnish blend) even though they generally change the color of things. Just a thought.