NH requires all short term renters to apply for a licence 30 days prior to renting, NH form CD-3, and then to file by the 15th of each month using form DP-14 or electronically. Companies like Avalara make this process a little easier, and charge a $50 set up fee and $20/month to perform this process for you and pay the tax. They send you an email each month, and you just have to enter your taxable income for that month.
If the Dept of Rev finds you listing, they will request you to pay all taxes due on prior years. The fee is 10% for failure to file and 5% for failure to pay, plus interest. I sense they are searching the listing web sites and have actually heard there is a program they can use to scan these sites to see who offers rental in certain areas (they just did this in Newport, RI).
AIRBNB needs to add this fee, I have read that other states are able to add this fee, althought that sounds innacurate, becuase I can't imagine AIRBN would add programing for just one state when all can benefit. I list on other sites that allow the fee, and it is frustrating that AIRBNB doesn't, and have passed on the feedback to the site.