They do in fact exist, though they often use smaller helicopters.
A Google search on the term "heli taxi" yields thousands of results, and while a lot are probably irrelevant, the first few pages give hundreds of operators around the world, ranging from companies ferrying passengers between airports and major cities to companies servicing oil platforms at sea, islands too small to have airfields, etc.
aviation.stackexchange.com/users/900/jwenting
Air Greenland is a good example of a company that uses helicopters in scheduled air traffic.
Unlike many of the other examples mentioned here, a large part of Air Greenland's helicopter operations take place as scheduled traffic, and not charter flights.
They operate the following: 2x Sikorsky S-61N with 19 seats 9x Eurocopter AS 350 with 5 seats 8x Bell 212 with 8 seats to and from multiple small cities and towns on the coast of Greenland.
aviation.stackexchange.com/users/14957/expeditedescent
The other answers demonstrate that there are several helicopter airlines, but I don't think that really answers the question.
Why aren't there more?
Simple economics.
Compared to other kinds of powered aircraft, helicopters are small, slow and expensive to run.
They do have the advantage that a helipad takes up much less space than a runway which means you could fly your helicopters right into the middle of the city, rather than to large airports in the suburbs or beyond.
However, helicopters are also very noisy, so that would be very unpopular with everybody else who used that part of the city.
This means that your heliport actually has to be out in the suburbs anyway, and you've just lost your only advantage.
All of this makes it very difficult to be a profitable helicopter airline.
All of the examples in the other answers are cases where special circumstances mean that helicopters can still work.
Taking people to oilrigs by air is much faster than by ship, but you can't land a fixed-wing plane on an oil rig.
Small, widely spaced communities might not be accessible by road and might not have the resources to maintain an airport, but might be willing to pay enough to support a helicopter service when it's their only option.
If you just want to transport "10–20 people a short distance", a bus is going to be almost as fast, much more comfortable and orders of magnitude cheaper in almost all non-special circumstances.
aviation.stackexchange.com/users/946/David Richerby
We do: Helijet International .
Helijet International is a helicopter airline and charter service based in Vancouver, British Columbia, Canada.
They operate regularly scheduled flights between: Vancouver Harbour Heliport and Victoria Harbour Heliport Vancouver Harbour Heliport and Nanaimo Harbour Heliport (Monday-Friday) Vancouver International Airport and Victoria Harbour Heliport(Monday-Friday, limited flights)
I'm sure there must be other services where the geography and economy can support it.
aviation.stackexchange.com/users/17559/Spaig87
There are helicopter airlines.
Definition: airline is a company that transports people by air and in so doing
makes a profit (or tries to).
PHI , Bristow , and CHC are three of numerous companies who transport people to and from oil platforms as passengers, for a profit.
(At least, that's their aim).
aviation.stackexchange.com/users/17262/KorvinStarmast
Erickson Aviation operates a regular helicopter service from Nome, Alaska to Little Diomede Island, Alaska.
Little Diomede sometimes makes it onto lists of the most remote towns in the USA and is more or less a small rock in the middle of the Bering Strait with a fishing village on it.
aviation.stackexchange.com/users/17532/Robert Columbia
New York Airways used to operate a helicopter taxi service until they closed in 1979.
They operated a number of helicopters, including the tandem rotor BV107-II.
If you ever watch the Clint Eastwood film Coogan's Bluff (1968), you'll see at least two of them operating off the Pan-Am building.
New York Airways
aviation.stackexchange.com/users/17576/Nathan
The helicopter option is popular with Hong Kong's high rollers, looking to hit the tables in Macau.
http://www.skyshuttlehk.com/
aviation.stackexchange.com/users/17611/William Avery
In the UK, British International Helicopters (formerly British Airways Helicopters ) is a helicopter airline, although they do not currently fly any regular public routes.
BIH fly for offshore (e.g. North Sea oil) and defence industries.
They also offer training and sightseeing tours from London (Redhill) and Coventry.
BIH operated a regular Sikorsky S-61 service from Penzance Heliport to St Mary's Airport and Tresco Heliport on the Isles of Scilly from 1964 until 2012, when Penzance Heliport was sold to the supermarket Tesco.
There are plans to reinstate the service if a suitable new site can be found.
BIH jointly operated the Airlink shuttle service between Gatwick and Heathrow between 1978 and 1986.
The service ended because its licence was revoked after the M25 motorway had been completed.
aviation.stackexchange.com/users/14762/keith jackson
Just a curious thought. Why isn't there any helicopter airline? Airlines that operate large helicopters that can transport 10 or 20 people a short distance?
Why don't we have helicopter airlines?
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<airline-operations><helicopter>
aviation.stackexchange.com/questions/32428
aviation.stackexchange.com/users/9126/TBBT
Helicopter airlines do exist. Some examples include Air Greenland, New York Airways. However, helicopters are small, slow, expensive and noisy which explains why they are not used more widely. In fact, there exist many helicopter airlines like Helijet, Erickson Aviation, British International Helicopters, Air Greenland and, formerly, New York Airways. They mainly connect remote locations and service oil and defense industries or high profile customers, although their helicopters are usually smaller. They are not so common because it is not profitable and helicopters make a lot of noise, so they are not preferred.
In fact, there exist many helicopter airlines like Helijet, Erickson Aviation, British International Helicopters, Air Greenland and, formerly, New York Airways. They mainly connect remote locations and service oil and defense industries or high profile customers, although their helicopters are usually smaller. They are not so common because it is not profitable. Helicopters make a lot of noise, so they are not preferred.