National STEM Education Tax Incentive for Teachers Act of 2013 - Amends the Internal Revenue Code to allow certain full-time elementary and secondary school teachers of math, science, engineering, or technology courses a refundable tax credit for 10% of their undergraduate tuition up to $1,000 in any taxable year. Increases such credit amount to $1,500 for teachers in schools serving disadvantaged children. Terminates such credit after the 10th taxable year for which such credit is allowed.