I understand the question as asking :
Are there universal or partial conditions/results that if true we know
that the conditional expectation (which is the best predictor in mean-squared error sense) will be a linear function of the
Then, apart from the well known Multivariate Normal Distribution case (where the condition is that all variables involved follow jointly a mutlivariate normal), we find some results in Johnson, N. L., Kotz, S., & Balakrishnan, N. (2002). Continuous multivariate distributions, volume 1, models and applications (2nd ed) . p.6-10.
Specifically in p. 9, there is a table(44.1) of bivariate distributions whose resulting conditional expectation (of the one variable on the other) is linear in the conditioning variable. All these bivariate distributions belong to the Pearson system (it requires some familiarity with the Pearson System to understand which "named" distributions are included in the table). The authors say that most of these are treated in detail later in their book.
Also, at the end of p.8 the authors mention the work of Steyn, H. S. (1960, January). On regression properties of multivariate probability functions of Pearson’s types. In Indagationes Mathematicae (Proceedings) (Vol. 63, pp. 302-311). who showed that for multivariate (not just bivariate) (Pearson) distributions, if the joint density vanishes at the extremes of the support of the conditioned variable, then the conditional expectation will also be linear (in all the other variables).
What I take as a lesson from the above, is that general results seem to exist only for mutlivariate distributions that give the same marginals (same family possibly different parameters) to all variables.