For unbounded continuous responses, Gaussian errors are the analyst's default model for many reasons, one of them being that their ML estimate coincides with the OLS estimate that has many desirable properties.
I am not aware of similar results for strictly positive continuous response variables, and I doubt they exist. Previous answers (and my own knowledge) seem to punt on this question, as in https://stats.stackexchange.com/a/79517/36229 and https://stats.stackexchange.com/a/136850/36229.
In the second answer, for instance, it says that in the gamma model the variance is proportional to the squared mean, and to the cubed mean in the inverse Gaussian model. And if I'm not mistaken, the lognormal model also has the variance proportional to the squared mean, with the additional interpretive benefit of being the logarithm of a Gaussian.
In the absence of a theoretical basis for one or the other, what should be my go-to model, and why? Or, as per whuber's suggestion in the comments, is there a principled way to select one or the other?
Or, is the difference small in practice and the one I choose is a matter of convenience and interpretive relevance (as with logit vs probit)?
And what about link functions? The canonical gamma link is the inverse mean, but I usually see the log mean used in practice. For argument's sake, let's assume we're using a log link. As per the comments, the link function is probably more important than the error distribution itself, but it's also a bigger/different question. Not to mention the fact that log links are used with all of the gamma, lognormal, and inverse Gaussian distributions. And inverse links are tough to wrap your head around.
Edit: for what it's worth, I found some guidance here and here that use simulations, and recommend the gamma distribution overall. Glen_b demonstrates that the log link tends to over-correct gamma errors, inducing left skew rather than symmetry but, like many others, seems equivocal overall on which one to use when. Firth (1988) shows that the gamma is slightly more efficient under certain kinds of misspecification.
Edit 2: You could always fit both and choose one based on some goodness-of-fit criterion, but in my opinion that's neither principled nor satisfying.