The road will connect the A5 at Welshpool Road roundabout in the west to the Ellesmere Road roundabout in the north, with new bridges over the River Severn and the Shrewsbury to Chester railway line.
The 282-page business case for the road states it will have a "moderate to slight" adverse impact on buildings.
The society says the buildings and streets should be in an environment free of noise, visual intrusion, air pollution and road traffic danger.
Byron Grainger-Jones, chairman of Shrewsbury Civic Society, said: "We have a clear remit to protect, promote and advance the historic buildings and street layout of the town of Shrewsbury.
"It follows that the civic society would wish to encourage the context within which these buildings and streets can be experienced and enjoyed by visitors and residents alike and this context includes a commitment to reduce the visual intrusion of vehicles, especially larger vehicles and to reduce noise, air pollution and road traffic danger.
"If we wish to celebrate and enjoy the wonderful buildings and delightful streets we must take an interest in the ways that environmental quality encourages or discourages the peaceful enjoyment of our heritage.
"Our interest in the North West Relief Road is based on the principle of peaceful enjoyment of heritage, listed buildings and medieval streets and this requires interventions that have a proven record for reducing noise, pollution and road turf danger.
"Building new roads in, around or the on edge of urban areas increases traffic levels, a result often referred to as 'new roads generate new traffic' or 'induced traffic'.
"The North West Relief Road will generate new traffic and this will damage the degree to which visitors and residents can enjoy the listed buildings and streets in an environment free of noise, visual intrusion, air pollution and road traffic danger."