It's not possible for an electron to emit or absorb a photon without the presence of a third particle such as an atomic nucleus; without the third particle, it's impossible for such a process to conserve energy and momentum.
However, if tachyons exist and couple to matter, then a material particle can emit or absorb tachyons while conserving energy and momentum. According to an interpretation originated by Bilaniuk (1962), inspired by the Feynman-Stueckelberg interpretation of antiparticles, tachyons are always taken to have positive energy, but this implies that an event that one observer sees as an absorption can be seen by another observer as an emission. For instance, it's possible for a moving material particle to spontaneously emit a tachyon, but in the particle's rest frame this would be seen as absorption.
Spontaneous emission is hard to make sense of in a classical theory. In a quantum-mechanical theory, it would be analogous to radioactive decay. This decay would have to occur with some rate. We normally expect a radioactive decay to occur at some fixed rate in the parent's rest frame, and this rate is lowered by the Lorentz factor $\gamma$ in any other frame.
What seems suspect to me about the idea of spontaneous tachyon emission is that there seems to be no way to reconcile it with Lorentz invariance. Let's say it occurs with some mean lifetime $\tau$ in a certain frame, in which the parent particle is moving with some speed and has some value of $\gamma$. Lorentz invariance seems to require that in the particle's rest frame, the lifetime should be $\tau/\gamma$. But in the particle's rest frame, the process is absorption rather than emission, and it can't have some fixed rate. The rate has to be determined by how many tachyons are available in the environment to be absorbed.
My question is whether my interpretation is right, and whether it constitutes a problem for Bilaniuk's claim that his approach eliminates all the paradoxes associated with tachyons. (I'm also pretty suspicious of his claimed resolution of the Tolman antitelephone paradox, but that's a different topic.)
Bilaniuk, Deshpande, and Sudarshan, Am. J. Phys. 30, 718 (1962). For an exposition of the ideas, see Bilaniuk and Sudarshan, Phys. Today 22,43 (1969), available online at http://wildcard.ph.utexas.edu/~sudarshan/publications.htm .