Shropshire Council's northern planning committee has voted to agree the proposals for Rowley's House and Mansion.
The Grade II* listed timber building is located off Barker Street in the town centre and dates back to the 1500s.
In recent years parts of the building have been surrounded by scaffolding to protect sections which need repair work.
The council application asked for listed building consent to replace and repair panels, as well as some parts of the timber frame of the building.
The proposal was unanimously approved by councillors on Tuesday, with committee member Councillor Ted Clarke welcoming the prospect of the scaffolding being removed.
"I am really pleased and will be supporting the proposal," said Councillor Clarke.
He added that the building was particularly symbolic for the town, and had previously been used to advertise it to visitors across the country.
Councillor Clarke explained: "Rowley's House figured on an old historic railway poster that was on rail waiting rooms across the country, inviting people to come to Shrewsbury – Rowley's House was the focus."
Shrewsbury Civic Society had welcomed the plans – and the potential for the scaffolding surrounding the building to be removed.
In a statement submitted in response to the application, it said: "Shrewsbury Civic Society is pleased to see progress being made on the restoration and repair of one of Shrewsbury's most iconic buildings.
"We are also pleased to note that it is the intention to remove at least two of the scaffolding towers as soon as possible."