The intersection of differential privacy (DP) and causal inference is crucial for protecting data privacy while preserving the accuracy of causal estimates, yet limited research explores how DP mechanisms impact causal effects estimated from privatized data. This paper formally investigates for the first time the impact of DP on causal inference, focusing on how standard DP mechanisms affect the accuracy of treatment effect estimates across various causal inference frameworks. Our core theoretical findings reveal that while applying DP mechanisms to outcomes preserves the unbiasedness of average treatment effect (ATE) estimates, it can increase their variance and severely distort individual treatment effect (ITE) estimates. Privatizing treatments, on the other hand, can lead to unexpected consequences, such as ITE underestimation in certain settings. Informed by our theoretical analysis, we propose two solutions to these issues: First, we show that under some conditions with privatized treatments, we can exactly recover the ATE estimates given information about the true privacy and balance parameters.Second, we propose robust regression as a mitigation strategy during the data analysis stage, to reduce the ITE estimation error. In addition to providing actionable guidance for balancing data privacy and causal inference accuracy, this work provides the first foundational results on treatment effect estimation from differentially private data, laying the foundation for future research into privacy-aware causal representation learning toward enabling models to robustly handle privatized data.