Given a regression based model with many feature variables; what tools would you utilize to figure out which feature variables add the most variance?
There might be a smarter method but I would simply try to fit a model without $X_i$ for every feature $X_i$ (and also a reference model with all the features). By contrast the model where $X_{49}$ is removed should obtain the lowest variance if $X_{49}$ is responsible for a lot of variance.
Be careful that in general a feature which causes a lot of variance is an important one, since if it wasn't important then it wouldn't have much impact on the target.