I know that omitted variable bias isn't a major problem in forecasting, but are other endogeneity issues (such as simultaneity or measurement error) going to be a problem if I am only interested in forecasting?
Its sure that endogeneity is not acceptable thing if our goal is to find structural/causal effect. You are focused on forecasting then endogeneity, as produced by omitted variables, actually don't is a major problem. Endogeneity produce, first of all, biased parameters estimates. Other sources of endogeneity as, measurement errors or simultaneity/reverse causation, produce biased parameters estimation as well. However if your goal is forecasting (or contemporaneous prediction as well) your major problem is overfitting. This concept is related to loss function as mean square errors, that you have to minimize, and appear when we consider in sample vs out of sample measure.
Key concept for understand the crucial distinction in argument is bias-variance trade off. Read my explanation here (Are inconsistent estimators ever preferable?) and, mostly, the cited article to which it refers to.
For another explanation you can read this article http://statisticalhorizons.com/prediction-vs-causation-in-regression-analysis
EDIT: I embraced the distinction between causation and prediction in light of the arguments contained in Shmueli (2010), mainly based on the bias-variance tradeoff. The bias is not the core but it play a role in prediction also. Therefore the “theory” play its role in prediction as well. Then, the so called “data driven” (correlational driven) model can be seen as too extreme perspective even if our goal is pure prediction; the magnitude of bias matters. However this magnitude depends from the "true model" and in any real situation it is unknown; so the magnitude of bias. Fortunately this problem is only theoretical and, at least in my opinion, irrelevant. Indeed the relevant thing is that the bias-variance tradeoff give us a justification for see the regression in two markedly different way and, more important yet, give us a justification for develop very different metrics to adopt. Infact the perspective about regression in causal inference and in predictive learning are markedly different. Moreover, also more relevant difference exist in the tools/metrics commonly used therein. If we do not accept a clear separation between causation and prediction those difference in regression practice are very hard to justify.
For example, models like ARMA and ANNet are “free of theory” by definition, they are purely correlational driven (data driven). The growing area of predictive learning, as a whole, follow the same perspective. Those models has demonstrate their effectiveness in practice and their superiority for forecasting purpose in comparison to structural models. While structural models is a necessity for causal inference. Latinus ancient people said in medio stat virtus; however, in my experience about causation vs prediction story, in the middle I see only confusion.