I notice it, but some people don't seem to. On a bike like this that turns in really nimble-like, I am pretty sure I'd notice it. Any added weight to the wheels (heavier tires in my case) is felt in the form of fighting a gyroscope.
When I put a rear tire on the front of my TW200 I immediately noticed that to lean it over in a turn it felt like you had to 'break it free' from wanting to be naturally upright. And once it broke, it felt like it fell rather than a natural smooth lean down. So I immediately bought another new set of OEM tires which solved my issue. The weight difference between the tire that was bugging me and the one that worked perfectly was four pounds. (8 pound tire vs 12 pound tire)
From what I read, the spoked wheels weigh more than that difference over the aluminum ones. Like anything, I imagine if you don't know differently, you just get used to it and that's what ends up feeling natural and normal.