I found something similar doing a quick search for flights between Akron-Canton Airport and Cleveland Hopkins. Seems you can’t do it without a layover somewhere else far afield from either.
I was flying to Phoenix via SLC. Coincidently, my stepson later had to schedule a flight to LAX via SLC and got on the same flight I did. Because of the time I bought my ticket vs. the time he bought his it was cheaper for me to fly from Colorado Springs to DIA while it was cheaper for him to drive just so we both could get from the Springs 73.05 miles to Denver for the same flight.
Topic Air in Belize has a 5 minute flight from Ambergris Caye to Caye Caulker.
The shortest one I’ve ever been on personally was Indianapolis-Chicago. It’s listed at 55 minutes but is actually more like 25.
Once upon a time you could actually fly between Dallas and Ft. Worth.
Ketchikan to Metlakatla is 13 minutes
I once took a flight between neighboring cities that was barely up in the air before the descent phase started.
I Googled to see what the current flight times are. 4+ hours. No more direct flights.
Isn’t the hub system wonderful?
I’ve got a candidate for shortest flight/biggest plane:
I was flying to Kinshasha, DRC and was on a 757. It first landed in Brazzaville, RC, then took off across the Congo River and landed in Kinshasha. It was maybe a leisurely 15 minute flight. I can’t imagine how much fuel was burned getting that thing in the air just to land it again.
There used to be flights from one side of Houston to the other, between Houston Intercontinental and Ellington Field, about 30 miles apart.
There used to be flights from Columbus to Dayton too. I think there was a regional carrier for US Airways that had a small hub there.
The shortest flight I’ve personally been on was a USAir DC-9 from Ithaca to Elmira, New York. That’s 32 air miles. I distinctly remember the pilot saying airtime would be 8 minutes.
The flight continued to Pittsburgh, where I made a connection to San Francisco.
According to Wikipedia, there used to be scheduled flights between MSP and St. Paul’s downtown airport. It doesn’t say the flight time, but the straight-line path is 7.5 miles, so I’m guessing right around 5 minutes, mostly because the runways are perpendicular.
I have flown from Mahe to Praslin Island in the Seychelles which is about 10 minutes.
There is small plane short and then there is big plane short. The most big plane short route I know of in the U.S. is St. Thomas to St. Croix, USVI. You can take a large jet from the U.S. mainland, land in St. Thomas, let people out and then continue on to St. Croix. The distance is only a little over 40 miles and takes a few minutes. You never climb to altitude. The runways at the airports involved are aligned so that you just take off and immediately start your final approach into St. Croix. Total flying time is well under 10 minutes.
The shortest major route that I know of in the lower 48 is Denver to Colorado Springs or vice versa. They are about 80 miles apart and both have large airports. Total flying time is about 12 - 15 minutes on the larger planes that fly between them.
The shortest flight I’ve ever flown was the twenty mile flight from Plattsburgh, NY to Burlington, Vt.
If you’re in New Zealand
Picton to Wellington is 25 minutes
Nelson to Wellington is 35 minutes
South Korea, which is about the size of Indiana, has nine commercial airports and eleven civil airfields. Gimpo to Incheon (both serve Seoul) would be pretty short and might exist as a layover, but since both airports are on the same subway line, I doubt there’s much demand. There are flights available between Daegu and Busan and it’s a short flight.
There is an AirTran flight between Columbus and Dayton, OH listed as 6 hours 25 minutes travel time, for $712 round trip.
Apparently it’s by way of Atlanta.
You could probably bicycle there faster.
Chattanooga, TN to Atlanta, GA - scheduled times are around 45 minutes. From experience, actual flight time is typically around 20 minutes - taxiing/deplaning in Atlanta take longer than flying.
That’s the shortest commercial flight I’ve been on. We just reached cruising altitude before starting our descent. But they did serve a snack. 
Looking at the information available from various tables at OpenFlights.org.
Taking it as a challenge to automate some stuff, I’ve looked at all the scheduled routes that start at airports with codes that begin with A or B. So far, have found nine routes that are 100km or less. No idea the nature of these scheduled routes, but might provide some candidates.
Alaska seems a good pool of them, but they have lots of towns wit no connecting roads.
Atmautluak, AK to Bethel, AK - 25.36km
Batsfjord, Norway to Berlevag, Norway - 38.44km
Belmopan, Belize to Belize City, Belize - 58.20km
Allakaket, AK to Bettles, AK - 62.32km
Alliance, NE to Scotts Bluff, NE - 68.55km
Kodiak, AK to Amook, AK - 84.94km
Atqasuk, AK to Barrow, AK - 89.76
Al Ain, UAE to Abu Dhabi - UAE - 99.30km
Aalborg, Denmark to Arhus, Denmark - 99.89km
Not a record, but there are daily direct flights between Skagway and Haines, Alaska, that are scheduled at 15 minutes. It is possible in theory to drive between those towns, which are 16 miles apart as the crow flies. Trouble is that it’s an eight-hour trip by way of Whitehorse, YT. There’s also a ferry.