In the relativistic context the answer to the question in the title of this thread is clearly yes. In the electromagnetic context it's known as "radiation reaction" and a notorious problem if you consider point particles. It's not fully consistently solved yet, and I think it's not necessary to fully solve it (if this is possible at all, which I pretty much doubt, but which I also cannot prove mathematically), because the classical point-particle picture is not applicable anyway, because quantum effects have to be taken into account, and there the situation is much better, i.e., QED is well-defined in the perturbative sense at least as an effective theory for not too high energies.
All this analogously holds for gravity too. There you have gravitational waves which affect the motion of the sources (as, e.g., the famous Hulse-Taylor pulsar demonstrates with high precision). Of course in this case we don't have a working quantum description yet.