I'm not sure I'd call Lignum Vitae "oily". "Resinous" might be a better term. LV is renowned for its high resin content (that's what makes it so dense it sinks). When you heat Lignum Vitae (e.g. with friction), the resin boils up. It actually makes a decent lubricant which is one of the reasons LV was used for propeller shaft bushings. The resin, however, does not evaporate away.
That is a crucial difference between Lignum Vitate and "oily" woods like rosewood or teak. Over time, in sunlight and low humidity, those woods will dry out. It's particularly noticeable in veneer or other thin applications - such as pens. That is why I have to reapply teak oil to my danish modern furniture every year.
Therefore, I don't try to rob wood of its natural oil for a finish - I supplement it instead. Teak Oil doesn't seem to darken wood nearly as much as Tung Oil or Linseed Oil, so I use it on many teak and rosewood family (and similar) pens. It yields a no-gloss natural-wood-feel finish that will probably need to be replenished from time to time.
Regards,
Eric