The most obvious experimental signature of tachyons would be motion at speeds greater than $c$. Negative results were reported by Murthy and later in 1988 by Clay, who studied showers of particles created in the earth's atmosphere by cosmic rays, looking for precursor particles that arrived before the first gamma rays. One could also look for particles with spacelike energy-momentum vectors. Alvager and Erman, in a 1965 experiment, studied the radioactive decay of thulium-170, and found that no such particles were emitted at the level of 1 per 10,000 decays.
Some subatomic particles, such as dark matter and neutrinos, don't interact strongly with matter, and are therefore difficult to detect directly. It's possible that tachyons exist but don't interact strongly with matter, in which case they would not have been detectable in the experiments described above. In this scenario, it might still be possible to infer their existence indirectly through missing energy-momentum in nuclear reactions. This is how the neutrino was first discovered. An accelerator experiment by Baltay in 1970 searched for reactions in which the missing energy-momentum was spacelike, and found no such events. They put an upper limit of 1 in 1,000 on the probability of such reactions under their experimental conditions.
For a long time after the discovery of the neutrino, very little was known about its mass, so it was consistent with the experimental evidence to imagine that one or more species of neutrinos were tachyons, and Chodos et al. made such speculations in 1985. In a 2011 experiment at CERN, neutrinos were believed to have been seen moving at a speed slightly greater than $c$. The experiment turned out to be a mistake, but if it had been correct, then it would have proved that neutrinos were tachyons. An experiment called KATRIN, currently nearing the start of operation at Karlsruhe, will provide the first direct measurement of the mass of the neutrino, by measuring very precisely the missing energy-momentum in the decay of hydrogen-3.
References
Alvager and Kreisler, "Quest for Faster-Than-Light Particles," Phys. Rev. 171 (1968) 1357, doi:10.1103/PhysRev.171.1357, https://sci-hub.tw/10.1103/PhysRev.171.1357
Baltay, C., G. Feinberg, N. Yeh, and R. Linsker, 1970: Search for uncharged faster-than-light particles. Phys. Rev. D, 1, 759-770, doi:10.1103/PhysRevD.1.759, https://sci-hub.tw/10.1103/PhysRevD.1.759
Chodos and Kostelecky, "Nuclear Null Tests for Spacelike Neutrinos," https://arxiv.org/abs/hep-ph/9409404
Clay, A search for tachyons in cosmic ray showers, http://adsabs.harvard.edu/full/1988AuJPh..41...93C Australian Journal of Physics (ISSN 0004-9506), vol. 41, no. 1, 1988, p. 93-99.
Yes, parts of a wave function can travel faster than light, but from my understanding, much of it has to do with the uncertainty of the position of the particle the wavefunction represented in the first place.
For example, there is active research into how to interpret the results of quantum tunneling experiments that indicate "superluminal tunneling." This recent article from Quanta Magazine‘ explains that research area well. There are several competing definitions on the tunneling time, because time duration is not an quantum observable.
It is thought, but as far as I know not proven, that attempting to use this vanishingly small superluminal part of the wavefunction to send information will always be less efficient than sending the light directly, because for a large barrier, nearly all of the wavefunction is reflected.
(I don't know about how to reason about the case Feynman described, because not enough context about the quote is given.)
Best Answer
A tachyon is a particle with an imaginary rest mass. This however does not mean it "travels" faster than light, nor that there's any conflict between their existence and the special theory of relativity.
The main idea here is that the typical intuition we have about particles -- them being billiard ball-like objects -- utterly fails in the quantum world. It turns out that the correct classical limit for quantum fields in many situations is classical fields rather than point particles, and so you must solve the field equations for a field with imaginary mass and see what happens rather than just naively assume the velocity will turn out to be faster than light.
The mathematical details are a bit technical so I'll just refer to Baez's excellent page if you're interested ( http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html ), but the conclusion can be stated very simply. There's two types of "disturbances" you can make in a tachyon field:
1) Nonlocal disturbances which can be poetically termed "faster than light" but which do not really represent faster than light propagation since they are nonlocal in the first place. In other words, you can't make a nonlocal disturbance in a finite sized laboratory, send it off to your friend in the andromeda galaxy and have them read the message in less time than it would take for light to get there. No, you could at best make a nonlocal disturbance that is as big as your laboratory, and to set that up you need to send a bunch of slower than light signals first. It's akin to telling all your friends all over the solar system to jump at exactly 12:00 am tomorrow: you'll see a nonlocal "disturbance" which cannot be used to send any information because you had to set it up beforehand.
2) Localized disturbances which travel slower than light. These are the only types of disturbances that could be used to send a message using the tachyon field, and they respect special relativity.
In particle physics the term "tachyon" is used to talk about unstable vacuum states. If you find a tachyon in the spectrum of your theory it means you're not sitting on the true vacuum, and that the theory is trying to "roll off" to a state of lower energy. This actual physical process is termed tachyon condensation and likely happened in the early universe when the electroweak theory was trying to find its ground state before the Higgs field acquired its present day value.
A good way to think about tachyons is to imagine hanging several pendulums on a clothesline, one after the other. If you disturb one of them, some amount of force will be transmitted from one pendulum to the next and you'll see a traveling disturbance on the clothesline. You'll be able to identify a "speed of light" for this system (which will really be the speed of sound in the string). Now you can make a "tachyon" in this system by flipping all the pendulums upside down: they'll be in a very unstable position, but that's precisely what a tachyon represents. Nevertheless, there's absolutely no way that you could send a signal down the clothesline faster than the "speed of light" in the system, even with this instability.
tl;dr: Careful consideration of tachyons makes them considerably different from science fiction expectations.
EDIT: As per jdlugosz's suggestion, I've included the link to Lenny Susskind's explanation.
http://youtu.be/gCyImLu0HSI?t=58m51s