New Hampshire does not have a general sales tax or an income tax on an individual's reported W-2 wages. There are taxes on an individual's interest and dividends income, business taxes, consumer excise taxes and other taxes for income earned in New Hampshire. See http://www.nh.gov/revenue/gti-rev.htm for details.
Your employer does not have to take out any taxes for the work in New Hampshire.
Unless you also do work for that employer in Vermont, that employer does not have to take out any Vermont state taxes; but you will have to include your income when you file your Vermont income tax return.
You should make estimated payments to the state of Vermont.
See http://www.state.vt.us/tax/pdf.word.excel/individual/estimatedpayments.pdf for more information, including the following statement:
"Estimated payments are generally necessary on income with no withholding or insufficient withholding. Making estimated payments is particularly important for residents working in another state or non-Vermont residents working in Vermont."