Last year, I built three raised beds and they were quite successful. But, at the end of the season when I cleaned them out, I noticed small, thin roots growing throughout, which, I assume, are coming from my neighbors fir trees that are on the other side of my fence. This week, I dug out all of the soil from one of the boxes and placed a barrier between the ground and the box (I realize this might only be a fix for a season or two). I want to shovel the soil back in, but my question is, do I have to remove all the small roots? I am removing as many by hand as possible, but some are so small and singular, I'm not sure what to do. Is it okay to leave them in there?
If you mean loose root pieces in the soil you want to put back in the beds, don't worry about small bits, just remove the larger more obvious ones, especially any in clumps. The small, broken root pieces are not going to grow, assuming they're not from some pernicious weed - from your description, they do sound like the fine roots put out by trees or large shrubs.
Are you sure they're not from your vegetable plants? If they were from a fir tree then I think that would be obvious - they would be connected to the tree by some more substantial roots.
I don't know what pernicious weeds you have in Oregon, but personally I'd remove on sight anything thistle or bindweed like (white, fairly uniformly 4mm thick and cord-like), dandelion, dock and cinqfoil (orange tapering taproot) or buttercup (tufts of relatively short white roots).