A) the data source for your website is a FileMaker database and
B) it's hosted on a FileMaker Server
you would have to look into server-side-scripting. In the FileMaker Server Admin Console you can create schedules that run as frequently as every minute, 24/7. These schedules can call scripts from within your FileMaker databases. The topic is too extensive, I can't provide step by step instructions here. But almost everything that you can do manually in a FileMaker client can be scripted server side with current versions of FileMaker/FileMaker Server (check server compatibility of individual script steps in your version of FileMaker). You can definitely send reminder emails this way, and you can definitely manipulate data in FileMaker fields, such as the status of a listing.
Thanks Michael... I appreciate the time you took to answer that so thoroughly.
So as far as scheduling a script is involved (assuming the Filemaker database source and server are as you stated) it might be along the lines of 'Check database records daily for listing's tenancy LEASE DATE FIRST and LEASE DATE LAST field', with an if > then statement requiring it to do any number of things such as alerting someone of an event and/or manipulate data in another field accordingly regarding it's status.... sound about right?
You might work with a find request that produces a found set of records which need action first (faster than looping through all of them, checking each one's fields with "if" statements). Once you have extracted those records which need action, you can then loop through them and perform any number of - server compatible - script steps.
For actions such as sending emails your routine might also need to include information on whether the action has already been executed for a particular record or not. You can then include that information in your search and avoid sending out emails multiple times.