That should be fairly simple and the device teardown shows you exactly what you need to do. I'll take you through it step by step.
Roku RC-MC1 remote Teardown - iFixit
Follow Step 2 to open up the shell. Now on step three you can skip the first picture of disconnecting the battery, instead go to the second photo that shows how to disconnect the microphone.
Unplug that and you've accomplished your goal! Now all you have to do is clip the shell back together and you can enjoy the peace of mind knowing it can no longer hear anything you have to say. If you feel ambitious or concerned you can remove the microphone altogether, but simply unplugging it will do the job.
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