My little experience here: I have an ancient chain saw which gives off a lot of bar oil (better too much than too little...). With it, I cut a huge block of beech out of a quite green tree in a long procedure, surely leaving some oil on the surface. After letting it dry for two years, I cut out some parts, again with the same saw, and after a couple of weeks planed the top and bottom with an electric hand planer.
I didn't even worry about oil residues. The noticable splatters after cutting all went away after time; probably the wood just soaks it in and it diffuses (and dried wood soaks oil really quick!). No traces are visible any more, anywhere on the surface -- neither the planed parts, nor where I let it rough.
OTOH, I have in the past used bar oil as an improvised substitute to treat tool handles (so, really rubbed them with a lot of oil). It does in fact leave a very nice finish, somewhere between a non-drying furniture oil and a drying oil. A bit tacky, but that's exactly what I want on an axe handle. Yet, I'd definitely go into them with tools, and have done so when refitting handles -- there's nothing strange there, it's just like any old tool handle with a patina.