Can a random forest be used for feature selection in multiple linear regression? Since RF can handle non-linearity but can't provide coefficients, would it be wise to use random forest to gather the most important features and then plug those features into a multiple linear regression model in order to obtain their coefficients? 
 A: The answer by @Sycorax is fantastic.  In addition to those fully described aspects of the problem related to model fit, there is another reason not to pursue a multi-step process such as running random forests, lasso, or elastic net to "learn" which features to feed to traditional regression.  Ordinary regression would not know about the penalization that properly went on during the development of the random forest or the other methods, and would fit unpenalized effects that are badly biased to appear too strong in predicting $Y$.  This would be no different than running stepwise variable selection and reporting the final model without taking into account how it arrived.
A: Despite the legitimate warnings that this approach might fail in some cases, this should not discourage you from trying it out! Breimann reports an example (Breimann 2001) that selecting features by variable importance from a random forest and plugging them into logistic regresission outperformed variable selections specifically tailored for logistic regression, and others report similar observations,  e.g., with using Boruta as a preprocessing variable selection step for logistic regression.
As both Random Forest variable importance computation and Boruta feature selection are readily available in R or other software and thus can be tested without much effort, this is something that should be given a try. If you have enough data, you can even validate the approach by doing both steps on different fractions of the data.
