Elmbridge Museum said its "museum without walls" initiative would provide its services in a "more accessible, relevant and cost-effective" way.
The last day to visit the gallery in Weybridge was on Saturday and the website was launched on Sunday.
Elmbridge council said Surrey County Council owned the museum building and would make a decision about its future.
The borough council said the new website offered a brand new interactive online experience that represented the museum as a whole and showcased its services to various audiences.
An "object in focus" section offers more information on selected objects and a "what we're up to" section looks at projects by the museum team, which is now based at the Civic Centre in Esher.
Actual objects will still be displayed at temporary exhibitions across the borough, and the museum will run a schools programme in September.
Two exhibitions are currently being held at Thames Ditton Library and Walton Library.
The museum's collection is being moved to a purpose-built store, where artefacts will be available to view by appointment.
A permanent museum gallery in Surrey has closed and its collection of 40,000 artefacts can now be viewed online.