Student loans can be forgiven after ten years of being a teacher (link). If there is no chance for a new teacher to pay off the loan in ten years, is there any benefit to pay anything other than the minimum (income-based) payment for ten years?
One obvious benefit is that you hedge against the possibility that for some reason you will not remain a teacher for the full ten years. You might decide you don't like it, or you might move somewhere where you're not able to find a teaching job, etc. If you paid only the minimum, you will have pushed the debt onto your future self.
Also note this on the page you link to:
Eligible Repayment Plans: Borrowers may use income-based repayment, income contingent repayment, standard repayment or a combination of these repayment plans. Payments made under other repayment plans (e.g., extended repayment and graduated repayment) do not count.
You should research what that means as far as a "minimum payment". It is possible (maybe even likely) that a payment plan that tries to game the system with bare-minimum payments (e.g., below the income-based repayment amount) will not be considered eligible.
If you're asking whether you should pay more than the income-based repayment amount, my impression is there's not much reason to (unless you're thinking you might not stay a teacher so won't get the 10-year forgiveness). The income-based repayment is designed to be a payment you can afford, so paying more than that might force you to downsize your lifestyle for no obvious benefit.