Hi Joumana

it is possible to apply visibility rules to tiles in the featured links block.

The visibility includes making this visible to an audience . 

If you included the audience rule  based on the user's last login date and select 'Before or on a date' that is one month in the future  and they login after a month in the future they won't be a member of the audience so the tile will not be visible to them.

Can you test this out and see if this works how you want.

regards