Some people living near to the listed building, which has recently opened as a visitor attraction and community venue after a multi-million pound restoration, fear an increase in anti-social behaviour if people are drinking there.
The Dye House is at the back of the main Flaxmill building and was built to help the mill produce dyed linen to compete with products from Ireland and Scotland. The visitor shop will be inside the main building.
Members of Shropshire Council's licensing sub-committee heard an application for the licence from Historic England.
Simon Cramer from Historic England said the shop would sell items for consumption off the premises.
"There will be local products, beer to link in the the malting use of the building and also local wines," he said.
Drinks would only be sold in the Dye House when the building was in use for a function, such as community theatre.
"It will not anything like the Flaxmill turning into a nightclub," he said.
Answering a question from Councillor Mike Isherwood, Mr Cramer said there had been alcohol on sale at the Flaxmill at events since its opening but said that had been covered by temporary licences.
"We have had not problems whatsoever," he said.
"We have a licensed officer on duty and fully-trained staff. The closing time will be 30 minutes after the licence finishes so people will not all be leaving at once."
Among the objections sent into Shropshire Council was one from a resident who said: "Our fears are that once given an alcohol licence they will then apply for an entertainment licence which again cannot be allowed when this building is so close to surrounding homes.
"The car park is not big enough to meet the Flaxmill buildings' needs and we will end up with our road being blocked by visitors' vehicles and the list goes on."
Another objector said: "We have already had to endure years of noise, disruption and suffering from the Flaxmill site in the hope that one day it will all be over and we can start to relax in our own homes again. If they are granted these licences, there will be no end to it and it will just get worse."
The licence granted will apply to the Dye House from 8am-10pm Monday to Saturday and 8am-8pm on Sundays, and the shop from 10am-5pm seven days a week.
A further application for a licence for a cafe at the Maltings will go before another licensing committee next month.