It depends on what the error estimates represent: errors in the measurements going into the model, or expected errors in predictions from a fitted model. For terminology, it's simplest to discuss in terms of the error variance estimates (which for a given study size bear a one-to-one relationship with the confidence-interval widths).
Standard linear regression, for example, assumes that the error variance is constant. If you have examined the distribution of measured outcome values and find that error variance actually depends on the values being measured, say with error increasing as the values increase, then you could consider a weighted linear regression to take that into account. In that case, you could be properly using information about observation variances to weight a model.
If you fit a standard linear regression, however, the estimated variance of predictions from the model are not constant across the range of predicted values even if the variances of observations going into the model and estimated variances of observed minus predicted values are constant. See this Wikipedia page for the formula for a one-predictor model. As the values of the predictor variable get farther from the mean of the original predictor-variable values, the variance of the predictions from the model necessarily increase. In that case the associated confidence intervals are best used to describe the result of the model; it would be inappropriate to use the variances of such predictions from the model to weight the results for a revised regression.
