I want to know if I can hide the values of a date field if the values are older than a certain time. The website is built with Drupal 7, and Display Suite.
The field formatter conditions module allows you to configure your field formatters with various conditions.
One of the included conditions is "Hide date field if from/to has expired", which should cover your requirements.
The module also has rules integration, but that shouldn't be needed for your requirements.
[EDIT] Actually it only allows working based on current time, not an arbitrary time.