Theoretically, it should be possible to land before taking off (if you go on a fast enough airplane between two cities close to a time-zone boundary).
Practically, are there any such flights, even if they were on opposite sides of the international date line?
I think just about every flight east across the international date line would qualify.
I'm interpreting this question as "Are there flights whose scheduled local arrival time is numerically earlier than scheduled local departure time?" All other interpretations appear to be silly(ier).
This will be satisfied by any flight of less than one hour duration which crosses a time zone boundary westbound. I thought of looking at flights into Chicago: it is near the edge of the Central time zone, but many nearby cities are in the Eastern time zone. I looked at a United Airlines timetable to find short flights from Chicago to destinations to the east, then looked up the corresponding westbound return flights.
The shortest such flight I found was United 5848 which departs South Bend, Indiana (SBN) at 4:33 PM EDT and arrives at Chicago O'Hare (ORD) at 4:20 PM CDT, a duration of 47 minutes. It only operates Saturdays. Recent FlightAware data suggests that SBN-ORD flights may have an actual flying time as short as 18 minutes. The airports are only 135 km (84 mi) apart.
There were other sub-hour flights from South Bend, as well as Muskegon, MI (MKG), but not as short as UA5848. Intriguingly, there are flights to Chicago from Fort Wayne, IN (FWA) and Grand Rapids, MI (GRR) scheduled for exactly one hour, so in terms of local time they arrive at the same time as they leave. An example is United 5345 which departs Fort Wayne at 4:25 PM EDT and arrives Chicago at 4:25 PM CDT.
American Airlines also has Chicago as a hub and had sub-hour flights from Fort Wayne and Kalamazoo, MI (AZO).
Yes, you can. And you can so so without an aircraft.
@Gagravarr's answer is closest to correct in real physical terms of any answer given.
The following is based on his deeply insightful response. I'm an engineer.
He is obviously [tm] correct in concept, but did not carry his point to its obvious conclusion. I do.
Non-engineers may find themselves querying the validity of this "solution" :-).
Time zones are artificial man made constructs which quantize the gradual variation of real local time as you progress around the globe. The time where you are at any moment IS the real time. To land before you took off you have to land at a place where the real time is earlier than the real time was when you took off.
As a given real clock-time travels around the earth with the rotational speed at that point, you need to travel faster than the rotational speed to decrease your current local time.
The earth rotates at about 1000 mph rotational velocity at the equator. So, at the equator, an aircraft travelling East at >= 1000 mph will have the real local time decrease as it flies. [I'll use 1000 mph as rotational speed at the equator to make the sums tidier. Actual speed is about 1040.4 mph and what this actually means gets a bit complex as discussed here Earth's Rotation - Wikipedia. 1000 mph is close enough for us.
However, that's at the equator.
As you leave the equator and travel pole-wards you can achieve the same effect at lower velocities.
At a lattitude of N degrees North (or South) the rotational velocity, and so the velocity required to "stop time" is
V_time-stop = 1000 x cos(latitude) mph (latitude in degrees).
So at 50 degrees South (bottom of NZ) V time_stop = 1000 x cos(50) = 640 mph.
At Severnaya Zemlya at 75 degrees North (near the site of the Tsar Bomba) V timestop = 260 mph.
At US McMudro base in Antarctica at 77.88S it's ~= 210 mph and
at Russian Vostok Station at 78.45S it's ~= 200 mph.
An operation Deepfreeze Hercules out of Christchurch on final into McMurdo IF flying due East will be experiencing decreasing real time if travelling at over about 210 mph groundspeed. (Add 4% if you want a slightly more accurate figure). The Russians (and anyone else) at Vostok have it just slightly easier.
Finally, at Amundsen-Scott station AT the South Pole if you walk the short distance to the actual pole, then
touch the marker with you RIGHT hand and
maintain contact while you jump along arcs of a circle around the pole marker
so that you rotate around the marker,
you will land some hours ahead of when you "took off' at every jump.
An added perspective. China operates a single time zone across the whole country - "Beijing time". In eg Urumqi or Kashgar the time is several hours later than it "should" be. Travel out of Western China to any surrounding country and as you cross the border you will "arrive" in the other country several hours before you left China - and no dateline involved.
Then there's Greenwich ... :-)
Not since Concorde stopped flying it hasn't.
British Airways' advertising slogan for a while for Concorde was Arrive before you leave. Between London or Paris and New York, you'd sit in a seat rather like this one, sip champagne, eat a little, and about 3 hours later arrive in Paris, "before" you'd left!
(For this answer, I'm defining "before" as the plane going faster than the revolution of the earth, allowing you to get ahead of the local solar time. This gets easier the closer to the poles you get!)
As others have said most any flight eastbound across the international date line lands before it takes off. Being an American married to a Chinese I've taken plenty of such flights. Once we even managed to take off and land AGAIN before the original takeoff.
However, it's also possible to do without the date line. I've done it once but it was so long ago I don't recall the city pair other than it's between capitals in Europe. ~45 minute flight that went one time zone west.
There's also another case that can produce a substantial jump. China operates all on one time zone but it really stretches across several time zones. A flight from western China to a neighbor would likely go back in time.
