If a jet engine is bolted to the equator, does the Earth speed up? If a jet engine is bolted to the equator near ground level and run with the exhaust pointing west, does the earth speed up, albeit imperceptibly? Or does the Earth's atmosphere absorb the energy of the exhaust, and transfer it back to the ground, canceling any effect?
 A: It's the latter. Look at the system earth + engine + atmosphere. Conservation of angular momentum must hold for the whole system (assume no gases leave the atmosphere due to the engine, which is a fair assumption).
A: Briefly, the Earth's rotation rate will change; and the net rotation rate of the atmosphere will change in the opposite way.  But very soon, friction between the air and the ground will ensure that the air and ground have no net rotation difference, which means that the Earth's rotation will return to the original rate.
A: Even if the device is able to eject gas out of the atmosphere (and I don't think such device, if existed, could be considered a jet engine), unless the gas reaches escape velocity in a high enough altitude so air viscosity is negligible, the gas jet will just follow a suborbital trajectory, falling right back into atmosphere, what would eventually cancel out the increase in angular momentum.
A 1 meter wide gas jet at sea level, even at escape velocity, would never make into upper atmosphere. All speed would be dissipated, first as huge sonic boom, and later as turbulence.
A: Addition to other (correct) answers:
Depends on the size of the engine. If it's large and powerful enough that it blasts parts of the atmosphere (and/or its propellant) into space, then you have a net force acting on the planet as a whole. Otherwise, no - as outlined in several other answers.
As for the feasability of such an engine... that's a different question altogether.
A: I think we can say "maybe, but even if it happened, there'd be no way to predict or prove it."
The other answers are correct, assuming a closed system. So can we find a mechanism (however silly) which could open the system, given a single, regular-sized jet engine?

*

*Premise: the butterfly effect means that even a small disruption of air could have a decisive affect on major weather formations, albeit in a chaotic and unpredictable fashion that we could not deliberately exploit.

*Premise: major weather formations affect the rotational speed of the planet: greater air density in the upper atmosphere slows the planet; greater density lower down speeds it up. Basic conservation of linear momentum.

*Premise: atmospheric escape happens when atoms and molecules in the upper atmosphere reach escape velocity: Earth loses about 3kg/sec this way.

*Hypothesis: when upper atmosphere mass is higher, atmospheric escape is greater. This seems a reasonable assumption, but I have no evidence for it.

If the hypothesis is true, then we have joined all the necessary dots: we've a mechanism for the jet to affect the weather; a mechanism for the weather to affect planetary rotation in the short term; and a mechanism for the short term effect to have permanent repercussions.
It'd be impossible to predict or prove what effect the jet had, if any; but it could have an effect.
A: Ok, so what will happen is that the rotation of earth will speed up, but negligibly.
On the other hand, the atmosphere will also rotate, but on the opposite direction. So, soon after the propulsion, the friction between the earth and the atmosphere will stop each othe and everything comes to normal.
A: Gedankenexperiment
Suppose you have a large torus floating freely in space.  The torus is sealed and full of liquid.  Inside the torus is a propeller (or screw, if you prefer).  I believe your question is reasonably equivalent to:

If the propeller in the torus starts spinning, will the torus experience angular acceleration?

Hopefully it is obvious that the propeller will accelerate the fluid in one direction, and that this will impart angular velocity to the torus in the other, thus causing it to spin [faster].  At the same time, once the propeller stops turning, you should be able to figure out why the higher angular velocity won't last: friction will eventually reduce any difference in angular velocities between the fluid and the torus to 0.
Obviously, the earth and atmosphere are not a torus or water, and the actual dynamics of the air is much more complicated.  However, this should make clear why the most common answer of: "It spins faster briefly, then slows back down again" is essentially correct.
On the other hand, if you want a more permanent increase, just leave the engine(s) running.  You could create a permanent "jet stream" at the level of your engine (much more effective if you have many engines placed around the equator).
A: A jet engine requires fuel.
Let us build it over a huge oilfield, next to a refinery. Over the years (centuries?) that it runs, the evacuation of the oil deposit will cause the Earth to lose mass and also to shrink slightly in diameter through subsidence. This will tend to speed up the rotation of the Earth.
There is no way to return all the exhaust gases back to the Earth and so the net angular momentum of the solid part of the planet will change permanently.
A: Total angular momentum with respect to the center of mass, in practice the center of Earth, is conserved. The plane will contribute angular momentum to the atmosphere and to Earth, totaling to zero. After a short while the air will give back this angular momentum to Earth and the original value is restored. In the mean time Earth has gained a tiny advance on its rotation, while the atmosphere has gained a tiny lag.
A: As other answers have stated there would be a small change in the Earth's rotation rate and a change in the opposite direction to the atmosphere's rotation rate. The size of the change would be limited by the increased friction between the earth and the atmosphere, and so the length of a day would settle to a new value if the jet engine were left running long enough. After the jet engine is turned off the increased friction would return the Earth to its usual rotation rate and the length of the day would revert to its natural value.
A: Yes, but it is so minuscule that it wouldn't be noticed. It's the same as if your friend was driving a car and you stick your head out of the window and blew in the opposite direction of the car's movement
A: Total angular momentum with respect to the center of mass, in practice the center of Earth, is conserved. When the airplane takes off, it acquires an angular momentum antiparallel to the Earths rotation axis. The total angular momentum of Earth plus atmosphere increases by the same amount. This angular momentum will eventually distribute over the entire planet. Long before this proces is completed the plane lands or crashes. Assuming it makes no turns its landing will decrease the earth + atmosphere AM back to its original value. If it turns while being up it again imparts angular momentum to the atmosphere. In the end it all adds up to zero but there are transient atmospheric effects.
