According to cancer.gov
Strong and consistent associations with cancers of the paranasal sinuses and nasal cavity have been observed both in studies of people whose occupations were associated with wood-dust exposure and in studies that directly estimated wood-dust exposure.
But there are no references on that page.
One 2015 meta-analysis says:
A total of 114 studies were identified and 70 were selected for review. Of these, 42 studies focused on the relationship between wood dust and nasal cancer (n = 22), lung cancer (n = 11), and other types of cancer (n = 9). Low-to-moderate quality evidence that wood dust acts as a carcinogen was obtained, and a stronger association between wood dust and nasal adenocarcinoma was observed. A lesser association between wood dust exposure and lung cancer was also observed. [...]
A meta-analysis that included four case-controls studies showed that workers exposed to wood dust exhibited higher rates of nasal adenocarcinoma than other workers (odds ratio = 10.28; 95% confidence interval: 5.92 and 17.85; P<0,0001), although a large degree of heterogeneity was found.
Although lung cancer is not the main type associated with wood dust, there's actually a meta-analysis on that too, which found more association with hard woods (or at least with the countries where more of that is processed):
Random-effects models were used to pool summary risk estimates. 85 publications were included in the meta-analysis. A significantly increased risk for developing lung cancer was observed among studies that directly assessed wood dust exposure (RR 1.21, 95% CI 1.05 to 1.39, n=33) and that assessed wood dust-related occupations (RR 1.15, 95% CI 1.07 to 1.23, n=59). In contrast, a reduced risk for lung cancer was observed among wood dust (RR 0.63, 95% CI 0.39 to 0.99, n=5) and occupation (RR 0.96, 95% CI 0.95 to 0.98, n=1) studies originating in Nordic countries, where softwood dust is the primary exposure. These results were independent of the presence of adjustment for smoking and exposure classification methods. [...] This meta-analysis provides strong evidence for an association between wood dust and lung cancer, which is critically influenced by the geographic region of the study. The reasons for this region-specific effect estimates remain to be clarified, but may suggest a differential effect for hardwood and softwood dusts.
But even in those Nordic countries where mostly softwood is processed, the association with nasal cancers does exist:
Studies of nasal cancer have shown increased risks associated with exposure to wood dust, both for those in furniture making and for those exposed exclusively to soft wood like the majority of Nordic woodworkers. We observed an SIR of 1.84 (1.66-2.04) in male and 1.88 (0.90-3.46) in female woodworkers.
I'm not sure of the oncological mechanisms, but one study found that even the cutting/sanding wood-processing operations release some (low) amounts of PAHs, known to be carcinogenic, even from "pure" woods, more from those with a high resin contents like oak. Interestingly the authors suggest that it's not just gaseous PAHs that are involved, but the heavier fraction which remains attached to the dust particles:
We have shown in our study the presence of PAHs in wood dust thus PAHs could potentially be “carried” by wood dust particles to where they deposit. Co-exposures to wood dust and PAHs therefore differ from co-exposures to wood dust and formaldehyde (a human carcinogen by IARC) in that PAHs are not just air borne (PAHs with less than five aromatic rings) as is the case for formaldehyde; but the dust acts as a vehicle for the heavier PAHs.
