As far as general relativity is concerned, the issue seems to date back at least to 1914, before general relativity was properly done, in Einstein's paper "Die formale Grundlage der allgemeinen Relativitätstheorie" (formal foundations of relativity). It includes the following statement on the topic :
An even deeper-reaching question of fundamental significance shall now be brought up - and I am not able to answer it. In the ordinary theory of relativity, every line that describes the movement of a material point, i.e., every line consisting only of time-like elements, is necessarily nonclosed, the reason being that such a line never contains elements for which $dx_4$ vanishes. An analogous statement cannot be claimed for the theory developed here. It is therefore a priori possible to imagine a point movement where the four-dimensional curve of the point is almost a closed one. In this case one and the same material point could exist in an arbitrarily small spacetime domain in several seemingly mutually independant representations. This runs counter to my physical imagination in the most vivid manner. However, I am not able to demonstrate that the occurence of such curve can be excluded from the theory that has been developed here.