How can I run a MySQL query that selects everything that is not null? It would be something like
SELECT * FROM schedule WHERE ((all)) IS NOT NULL
I've just looked at your linked question and agree with Guffa's answer that you should normalise your database.

The above looks more like a spreadsheet then something that belongs in an RDBMS.
To answer your concerns about this being the way you want it displayed. You could write a pivot query and put it in a view to mimic your current table structure and use this for your display query.
This will avoid the need for some horrific 24 column WHERE clause whenever you want to search for data or find out if it is NULL as in this question.