I just replaced a few two-prong outlets with GFCI outlets so we can plug in things built this millennium. Two of the outlets read "correct" on my circuit tester, but one reads "open ground." None of the outlets are grounded, so I don't know why this one would read differently. Anyway, does this matter? Should I bother to do something to fix it?
No. In fact, GFI protection is a safe and legal way to install 3-prong receptacles on non-grounded wiring.
You are likely reading a ground in some area because armored cable will show a ground with a tester but it is NOT a safe or acceptable means of ground in all cases. Only one type of AC cable will provide an acceptable ground.
None of the outlets are grounded, so I don't know why this one would read differently.
Most likely your first two outlets are grounded (though not necessarily in a reliable and code-compliant way). Even if there are no explicit ground wires boxes and devices can still be grounded through armored cables, metallic conduits etc.
Anyway, does this matter? Should I bother to do something to fix it?
As I understand it the National Electric code allows GFCI receptacles to be fitted without a ground connection in retrofit applications provided they are labelled with a "no equipment ground" sticker.
This is still a sub-optimal situation though, a GFCI with an open ground will NOT stop an appliance becoming live when a wire comes loose and touches the metal case. All a GFCI can do without any form of ground is limit the duration of the shock.
I presume that code allows this because it's the "lesser of two evils". If you assume people will use class 1 appliances on ungrounded outlets one way or another then encouraging them to incorporate GFCI protection makes sense.