iPhone and iPad suddenly can't access local IPs
It's looking likely that the problem is a bug in the DD-WRT firmware on my Netgear WNDR3700 router.  

I haven't been able to fix it yet.

People in that thread suggest that running the following commands in the router would be a suitable workaround, but I haven't gotten them to work yet:

swconfig dev switch0 set enable_vlan 1
swconfig dev switch0 set apply


UPDATE:
Oops, it's potentially not a bug in that firmware; I am even having problems after swapping out the router. But since this other router uses DD-WRT too (although a much earlier version of the firmware), I'm leaving this answer here since it's still my best guess and might help someone.