Originally Posted by Aeroplane
No, they won't. The width of the tires doesn't bring the rim any closer to the fork crown.
And the 4mm BB drop is really pretty damn negligible. So is the drop going to narrower tires (another 2mm? Maybe?). As long as you aren't trying to carve a super-wide slalom course with 175mm cranks you should be fine.
I agree that dropping your BB a few mms is not a big deal for normal riding, but it may become significant for an emergency maneuver. 4mm doesn't sound significant, but if you go with a 700x23mm, which is pretty common, that will put your BB about 1 cm lower than 27 x 1 1/8. To me that seems significant, but I would love for someone, given a certain crank length, q factor and pedal width, to calculate how many fewer degrees one could lean before pedal striking. I'd do it, but don't have the time to refresh my geometry skilz.