If an employee resigns but is also due a performance evaluation and merit raise before they leave, do we have to do both of these things?

It depends on more specific details of the situation and your company policy. Included below are some general tips and a specific example related to performance evaluations and merit raises.

Usually raises and performance evaluations are given at an employer's discretion; they aren't required but are sometimes offered as a form of incentive to employees. It would usually be considered a waste of resources to evaluate or give a raise to an employee who's already planning to leave, so you're justified in withholding those incentives.

Possible Exception

If your company's policy has a clearly defined schedule for mandatory performance reviews, and/or if your company has very specific prerequisites for earning a raise and completing those prerequisites automatically earns an employee their raise then you may want to grant those incentives even after an employee gives notice in order to avoid potential discrimination claims.

For example, let's say every employee is to be evaluated once a year before their date of hire anniversary, and part of the evaluation policy includes an automatic raise for employees who have consistently met the expectations for their job. If all of that's stated in your company's policy, then it would be a breach of policy and therefore has potential to be viewed as discrimination if you make an exception for any one employee, even if they just resigned.

Consider

Performance evaluations near the end of an employee's career with your company can be an excellent opportunity to show an employee how much you appreciate them. Especially if you'd prefer the employee not resign, offering a positive evaluation and a raise during their final days of work could be what it takes to change their mind.

Even if the employee ends up resigning, the fact that you gave them the performance evaluation and raise they were entitled to might act as an incentive for their peers.

Helpful Links:

Performance Reviews for Employee Changing Jobs or Supervisors

Effective Performance Reviews - HBR.org