I ran a specification with an outcome in $ terms, and a coefficient on a dummy variable was statistically significant. However, when I ran the exact same model with the outcome in log-dollar terms, the coefficient is not significant. Is this possible, or must there be a mistake?
That is perfectly possible. In my experience, this is often because of outliers, who tend to be less important when a log-transform is applied. Having said that, please observe that if you use logs, you are basically estimating a different quantity, so it can very well be that significance is lost even though there are no outliers whatsoever.