You mean if the house lost power, would the GFI trip -- mine never have.
I have FireShield GFI protected power strips. Fire Shield site We lose power occasionally, it never affects the fish tanks, they come back on when everything else does.
Now if it trips, like when I overfilled the tank and water got into the strip, then I had to dry it out and reset the thing, it wouldn't reset itself.
Most permanently installed GFCIs do not trip in power outages, so that the question does not apply. Many plug-in type GFIs/GFCIs do fail-safe after power outage as they are intended mostly for job/construction site use where this is a big safety factor. Those units should not be used in tank applications. Only the manufacturer can tell you if a plug-in type will fail-safe after power loss - but if the unit does not specify that it does not, assume that it does and that you do not want it for your tanks.
It would auto reset after a power outage that was not caused by a GFI trip. What I mean is if your house power went off and then back on, it would be still active. If you had something cause the GFI to trip off though, you have to manual reset it, this is a safety because it assumes you have a faulty piece of equipment which made it surge.
I looked at the packaging for this product in a local Lowe's (home improvement superstore), and it did not seem to indicate auto-reset capabilities. So I'm kind of conflicted.
FWIW I live in a rented apartment, so it's not really practical to have my outlets all switched to installed GFCIs.
If you are shoping at Lowes and have the sort of GFIC that is on a power cord, then it is likely the construction variety.
I suggest going to Walmart, in the electrical dept they usually carry the Fire Shield brand of power strip, this is the only variety of strip I've seen that is approved for aquarium usage. It has built in GFIC and 6 plugs and a switch. Quite reasonabley priced, at $12, you can get the one with surge protection also for about $24.
I also have used the plug in GFIC from Sears that is orange or yellow, I can't see it now, sort of hidden. It will not trip with power off like the construction types do. (I think I read once that the construction types will trip when power is off so that if someone has accidentally cut the extension cord that they are plugged into, then the whole thing will trip.