If my employee leases an apartment through the company and allows me to stay in the apartment during my workweek, do I have to pay payroll taxes?

Probably not.  If your employer is paying for the apartment, you are utilizing it as an alternative to a hotel for business travel (are away from home), are not using it for personal reasons or allowing a spouse or other companion to also use the space, then it sounds like a working condition fringe benefit.  Working condition fringe benefits are not taxable to you (See § 1.132-5 Working condition fringes).