This statement is a fundamental algebraic property of linear regression models (the omitted variable bias formula demonstrates that the bias of a coefficient depends on the covariance between the regressor and the omitted variable); therefore, no citation is needed.
The claim describes a core mathematical property of ordinary least squares (OLS) regression. According to the standard omitted variable bias formula, $\text{Bias}(\beta_1) = \beta_2 \text{cov}(x_1, x_2) / \text{var}(x_1)$. If the regressor $x_1$ is uncorrelated with the omitted variable $x_2$ (i.e., $\text{cov}(x_1, x_2) = 0$), the bias term vanishes. This is a matter of mathematical definition and everyday statistical observation, making it common knowledge requiring no empirical papers.