To me , they do , and a lot. Played the sh-6 with covers an uncovered on the same guitar and the difference was a bit more treble and attack. The covered one sounded more honky in the mids.
In my Les Paul I've had both uncovered and covered humbuckers (I soldered on the covers so they are the same pickups) and with the covers it cuts some of the highs and comes across as less aggressive, perhaps a bit darker as well. I'm preferring the covered tone to the uncovered tone since adding the covers about 8 months ago.
So, as a general rule of thumb, do pickup covers affect all pups in basically the same way, by cutting the highs ans focusing the mids and lows a little more?
My experience is limitted to SD's (59, JB and Distortion) with/without covers. Jind's description is accurate. The difference to me is fairly subtle and I'm not going to claim I have great ears for this, but I'd say that it seems more like a low-pass filter + a small drop in overall output when the covers are on. I'd call it "sweeter" sounding, but YMMV.
For what it's worth, I was looking at the new Dominions and noticed they sell a covered version with the poles visible. I'd like to try a set and compare since, personally I like the look of cover PU's better.
So, as a general rule of thumb, do pickup covers affect all pups in basically the same way, by cutting the highs ans focusing the mids and lows a little more?
Any guitar routed for a standard humbucker should support covers. Depending on the pickups you have, make sure you get the right spacing (F-Spaced, regular spaced, ...) if you have the screw head pole pieces.