I don't see how it could, since DHT _does_ go down, not up, when you take finasteride or dutasteride!
I suppose what you're really asking is, does the body sense the lower DHT levels and then take steps of some kind in an ATTEMPT to bring them back to "normal"? I think the answer here is kind of mixed...DHT has been shown to play a role in the feedback mechanism between the brain and the gonads which regulates the production of androgens (testosterone, to be specific). That's part of the reason why testosterone goes up a little bit with the use of finasteride/dutasteride: the brain sees less DHT floating around, so it signals the testes to increase the production of testosterone a little.
On the other hand, there was an
in vitro study a while back with human scalp hair follicles which measured the production of the 5a-reductase enzymes (both types) in response to androgenic stimulation (testosterone). They found that added amounts of testosterone INCREASED the production of the type 2 enzyme, but had no effect on the type 1 enzyme. That's kind of the OPPOSITE of what you'd expect, because if there were some kind of "balancing" mechanism involved here, you'd think that LESS androgenic stimulation would cause more of that enzyme to be synthesized.
So I think the bottom-line here is that no, I doubt that the body has any kind of serious mechanism in place to try to kick-up the production of DHT in response to finasteride. In fact, it may even go in the OTHER direction, at least as far as scalp hair follicles are concerned!
Bryan