Suppose a distribution has two parameters $\alpha$ and $\beta$, and let the maximum likelihood estimators for these parameters be $\hat{\alpha}$ and $\hat \beta$.
Is the maximum likelihood estimator for $\alpha-\beta$ necessarily equal to $\hat{\alpha}-\hat{\beta}?$
We cannot assume the independence of $\hat \alpha$ and $\hat \beta$ since they are both functions of the same data set.
Yes, the MLE of $\alpha - \beta$ is $\hat{\alpha} - \hat{\beta}$. It is the invariance property of the MLE, namely, the MLE of $g(\theta)$ is $g(\hat{\theta})$. The only requirement is measurability of the $g$. You can find the proof here.