It depends but if you have VT rental income and meet the VT nonresident filing requirements then you will need to file a VT nonresident state income tax return.
According to the VT Department of Revenue, if you were not a resident of Vermont for the full tax year but earned income in Vermont, then your income on rents and royalties from ownership of property located in Vermont will need to be included on a Vermont state income tax return as long as you are required to included this income as part of your federal adjusted gross income.
When calculating income from VT sources, if the result is net income of more than $100 or gross income (income before any losses) of more than $1,000, then you must file a Vermont income tax return.
You will also need to include this VT rental income on your NY resident state tax return. You will get a state income tax credit in NY for any VT state income taxes that you paid on your nonresident VT state income tax return.
You will want to work on your nonresident VT state income tax return first. You will then take a tax credit from your nonresident VT state income taxes on your resident NY state income tax return. (Please note that you will only get a tax credit for your VT state income taxes up to the amount of NY state income taxes that would have been paid if the income was earned in NY). The credit for taxes paid to another state section will be at the end of your residence state's interview process.
Just follow the TurboTax guide when working on your states (remembering to do your nonresident state return first) and TurboTax will do all the calculations and credits to your resident states return
Here is additional information about filing in multiple states (select "see more answer" to view the entire attachment)