Bryan has got it right.
To supplement:
Finasteride is a competitive inhinitor, wich means it competes for the same site (usually called the A- site, or the active site in pharmacological terms) of the enzyme as testosterone. Finasteride, having a greater affinity (wich on a molecular level means having a structure that better fits the shape of the A- site than testosterone), binds irreversibly to the enzyme and keeps testosterone from binding to it.
Thus, the enzyme is "inhibited". It is not destroyed or removed from the body in any sense, it is simply inhibited from having an effect on testosterone, because finasteride is "taking up space".
EDIT: after a while, it is, of course removed, as every protein in the body is metabolized and new ones are produced. Reason should state that when the inhibited enzyme is degraded, and a new one is transcibed, the new enzyme is NOT inhibited.
That is, unless, you keep taking finasteride.
One could speculate if finasteride has an effect on RNA transcription, or even DNA expression, of the 5A- reductase enzyme, but I am not familliar with any research on this. A qualified guess would be "probably not"