Is this sentence ambiguous?

What you’re getting at is whether the clause means:

Before the 6th of the month, Lessee will pay a penalty of $16.00 for rent that is unpaid.

or:

For rent that is unpaid before the 6th of the month, Lessee will pay a penalty of $16.00.

I interpret it with the second meaning on the basis of the structure:

Lessee will pay [penalty] for [violation].

To reach the first interpretation I’d expect it to be worded more like:

Lessee will pay a penalty of $16.00 for rent that is unpaid by the due date.

(And then continue with something like “From the 6th of the month the penalty will increase by $1.00 for each day the rent remains unpaid” to bring in the significance of the 6th of the month along with the additional provision that you mentioned.)

I think you’d need a contract lawyer to advise you whether it could be considered ambiguous in practice. And assuming you’re the lessee, it’s not in your interest to argue for the first interpretation anyway...