I rode for a while without the cover, and then put it back when I installed a DNA filter. No lid does offer a little more power in the lower RPM range than with it on and the OEM air filter installed. I really didn’t notice any greater induction noise, but I have a decat link pipe so my bike is slightly louder than stock.
There are a couple comments about it and a youtube link with dyno charts about it in this thread:
I have a coober ECU installed and both I and another forum poster talked to one of the techs at coober (very nice and informative guy) who recommends not pulling the air filter lid, as the bike already runs lean and he said it’s proven that it will damage (cook) the cylinders over time. I asked how long it takes for damage to start occurring and he said pretty much immediately.
I did it anyway. My bike didn’t blow up. But the DNA filter with lid on feels almost the same and I imagine is a wee bit safer in the long run.