Heterogeneous treatment effect (HTE) estimation is receiving increasing interest due to its important applications in fields such as healthcare, economics, and education. Current HTE estimation methods generally assume the existence of abundant observational data, though the acquisition of such data can be costly. In some real scenarios, it is easy to access the pre-treatment covariates and treatment assignments, but expensive to obtain the factual outcomes. To make HTE estimation more practical, in this paper, we examine the problem of estimating HTEs with a budget constraint on observational data, aiming to obtain accurate HTE estimates with limited costs. By deriving an informative generalization bound and connecting to active learning, we propose an effective and efficient method which is validated both theoretically and empirically.