Four tracks all the way would take a lot of buildings to be demolished; some of which are not that old.
Depending on what curve radii you are permitted, which would determine to what extent the line could "slalom" around buildings, I don't think it would be that many demolitions.
Almost everything east of Sackville Street is being demolished anyway, and there is easily room on the south side of the existing line, then there is a student-y bar that only really exists because of North CAmpus, a Starbucks and a multi story car park that is heavily used by uni people in the way, which are not particularly high value buildings. That gets you to Princess street, where if you can slalom to the north side of the tracks you might be able to get away with only demolishing the Joshua Brooks and the Starbucks building on Oxford Road to reach the station.
It would be a bit of a bendy line but not too bad, and once you are west of Oxford Road you could cantilever most of it over Whitworth Street until you reach Deansgate where you only have to demolish a handful of buildings, many of which are associated with the station.
It is far from a motorway through the city or something.