As best as I can tell, they don't need to "stay open" but they need to keep employees on the payroll for 8 weeks. If they lay off any employees, it reduces the forgiveness amount so it's advantageous to keep employees on the books even if business is closed due to not being essential or no customers.
edit: Business should lay off employees now before accepting the PPP (which is what I'm probably going to do). Ex. employer has 4 employees, they accept the PPP, they lay off 1 person, that's 25% payroll deduction for the forgiveness. Yikes.