What is the future of the airline industry when fossil fuels run out
in 50 to 100 years?

Fossil fuels might run out
- but the chances are that we'll find more

We are swimming in energy, tidal power, sunlight, even ambient heat
are sources of energy. Also the core of the Earth is ... well rather
hot.

Most likely airlines will not exist in about 30 years, either because
we are using vacuum tubes or 'sky hooks'
- either of which would be similar to replacing the bicycle with a railway train

When fossil fuels are not more avaliable, the amount of CO2 on
atmosphere will be more critical also. Even humanity discovers new
resources of fossil fuels We need to change the form of put the
engines to run.
The primary solution for the energy problem might be the use of
refinated vegetal oils. This solution will be demand a big work
arround the world in multiplicating vegetal species that can be used
to get such oils. For the other hand the emission of CO2 on atmosphere
will be controlated because we will fix carbon from atmosphere at same
rate than produced by consumption.
The big problem in this case will be the controling of the
environmental degradation produced by agriculture.
Anyways other resources like sun energy are avaliable to use when the
advance of technology permits an practical use of such type of energy.
Nuclear energy is an option, but not recomended for well knowed reasons.

The primordial obstacle to the advances in the alternative use of new
technologies is the marketplace lobby of the fossil fuels producers.

Sorry for the poor english.

While there are many alternative energy sources that are suitable for
ground based vehicles, airplanes have many influencing factors that
limit the practicallity of these technologies.  Batteries for instance
are incredibly heavy per equivalent energy density, and would weigh
too much.

Hydrogen is currently the most technically feasible alternative to
fossil fuels for use in airplanes, and is suitable for use in rotary
and jet engines.  Due to its incredibly high energy density, one
kilogram of compressed liquid hydrogen has as many calories as 2.8 to
2.9 kilos of jet fuel.

Hydrogen also burns much more efficiently than fossil fuels which emit
partially burned hydro-carbon chains. The synergistic effect of
vehicle weight reduction and more efficient engines could result in
dramatic increases in performance of airplanes (where weight is
especially important).

Perhaps the only reason we are not using hydrogen based planes (and
cars), is that it is not yet integrated into our infrastructure. 
Iceland is the world leader in developing a hydrogen economy where
they can both extract enough to meet their own energy requirements as
well as export hydrogen for profit.  The following links have
information on both the emerging hydrogen economy,  and about using
hydrogen in vehicles:

http://news.bbc.co.uk/1/hi/programmes/newsnight/archive/2208013.stm
http://www.bellona.no/en/energy/report_3-1999/11403.html
http://planetforlife.com/h2/h2vehicle.html

What is interesting from an environmental standpoint is that while
hydrogen engines do not release pollution (they only emit water),
steam released into the atmosphere at high altitudes may contribute
too a greenhouse effect.  I hope this was helpful.

It is funny you worry about airplanes when the oil ends. We will have
bigger worries if that happens.

But you could always have a nuclear--powered airplane for the president.