If a utensil can't become meat or milk through bliya, (like wood does) then is it ever possible to become non-kosher?
Indeed a utensil that doesn't absorb cannot become non-kosher.
Maran Rabbeinu Ovadia Yosef zt”l ruled in his works and Torah lectures in accordance with the opinion of Maran Ha’Shulchan Aruch that glassware does not require Hag’ala and it may be used for both meat and dairy. [...] According to the Sephardic custom, however, a thorough washing is sufficient, as we have explained.
However Ashkenazim consider that glass absorbs and has a status of Kli Cheres, which is to say that glass dishes are equal to that of pottery (which can become non-kosher, see e.g., here).
Your question mentions wood, but wood absorbs and is kashered through hagala (SA OC 451:8).