Does a college student pay federal or vermont state income taxes if earned under $6000 on w-2

It depends. Tax filing requirements depend on a few factors.

Federal:

• If you are not a dependent on your parents' return, and your filing status is anything but married filing separately, then you would not have to file. However, if you had federal taxes withheld, you may want to file to get a refund of federal income taxes withheld.
• If you are a dependent, you do not have to file with your wages, but if you had unearned income (interest, dividends) greater than $1,100, then you would have to file.

Although I provided general guidelines, there are many other situations that may be applicable to you on the federal level. Please see Do I Need to File a Tax Return? for IRS guidelines for your complete situation. 

Vermont:

Per the State of Vermont, you must file an income tax return in Vermont:

• if you are required to file a federal income tax return, and
• you earned or received more than $100 in Vermont income, or
• you earned or received gross income of more than $1,000 as a nonresident. 

Whether or not you are a resident or a non-resident depends on if Vermont is your "home state" or if you are a student attending college in Vermont. If you are away at college, you are a non-resident of Vermont and considered a resident of your home state.