What non-metric fastener heads are left on a standard bicycle? I believe that we've for the most part made the transition to metric on contemporary bicycles -- at least for the tools we use to repair them, am I right?

For example, while many screw threads are still imperial (or whitworth or some bizarre French/Italian/British concoction), the actual fastener heads or shanks are metric. For example, pedals are  9/16" x 20 tpi  but we've standardized on 15 mm for the pedal wrench (or a metric allen). Solid axles might be 9mm x 26 tpi but again we've standardize on 15mm for the nut.

Are there any parts of non-BSO bikes coming off the line where we might still expect to still need a non-metric tool to take a component off of?
Very rare, but I believe some pedals still require a 9/16" spanner to remove them. Not to be confused with the 9/16" axle like you have mentioned. Park Tool still make a pedal spanner with both a 15mm and 9/16" opening. Considering 9/16" of an inch is 14.3mm and the chances you'll actually have to remove one are fairly slim, I'd be happy with just a standard 15mm pedal spanner. 

Beyond this I can't think of anything else. Perhaps wheel nuts on very cheap kids character bikes? Some of those seem to come with whatever the factory had going spare at the time of assembly!