After following these steps, you will no longer be bothered by this again...
You have to do this through a web browser since these features do not seem to be available on the mobile Facebook app. First from the Home page look on the left side for the Apps section. Click the On This Day app. If this app is not shown, click the More link next to Apps, this will take you to a list of all apps you have and you'll be able to find On This Day there.
After On This Day is loaded, click Preferences.
In the Filters section, click Edit for Dates.
Click on Select Dates and then enter the earliest date that Facebook will allow you to and then manually enter a date some time in the future and click Save. It's better to make the future date quite far in the future.
Finally click Save and you're done.
You should stop seeing new memories after this point.