If all the stars disappeared simultaneously (from the earth's reference frame), how long would it take for the night sky to go completely dark to our unaided eyes? Would we still be able to see anything after a thousand years?
The farthest object that we can see with our naked eyes as of now is the Andromeda Galaxy (2.6 million light years away), although the M83 (14.7 million light years away) has also been reported to be seen with the naked eye. (You may want to click here for more information on what the farthest objects we can see with our naked eyes, from Universe Today or see this answer from this same site.)
Assuming that just before the moment all the stars disappear, these two farthest objects are still emitting light, then we can know judging by their distance from us, that this light will only reach our eyes 2.6m years later (because 1 light year = light travelling for 1 year) from the Andromeda Galaxy, and 14.7m years from M83.
Therefore to answer your question more directly:
We would still see the stars (assuming this is what you are referring to by "anything") for up to 2.6~14.7m years, depending on what really is the farthest thing we can see in the sky.
"The farthest star we can see with our naked eye is V762 Cas in Cassiopeia at 16,308 light-years away. Its brightness is magnitude 5.8 or just above the 6th magnitude limit." As per this information, if all the stars went out simultaneously, then it would take 16308 years for the sky to become completely dark to the naked eye. Of course, stars which are farther away will still be visible to powerful telescopes.
Depending upon how far a star is away from us, is exactly related to how long it would take for us to notice if it weren't there anymore.
Given your scenario, all stars instantaneously disappear, I will give a very short timeline.
At roughly 8 minuets and 25 seconds after the event, we notice the sun go dark.
After about 4 years and 146 days, $\alpha$ Centauri goes dark.
Sirius disappears in 8 years and 219 days.
$\varepsilon$ Eridani follows at 10 years 182 days.
M31 (Andromeda Galaxy) disappears at 2.5 million years.
I hope you see the pattern forming here.
The closer the star or other object, the sooner it will appear to go dark, or disappear. Therefore, I would assume that because every humans eyes are basically the same, roundish, with a roughly 5mm pupil. The most distant object is probably right around 2.9 million light years away, the galaxy M33.
Assuming that there will be no GRB within 30-60 seconds of all of the stars going dark, the sky would be completely dark to the unaided human eye in 2.9 million years. Not that humans will survive that long into the future, but that is a different problem all together.