Hi,
Modifying the basemap should not destroy your Map Tour.
But as Raymond pointed out, we added the ability to change the basemap from the builder to avoid mistake as yes it's possible to destroy a Map Tour if you modify your webmap in a non supported way (make the map tour layer not visible, add/remove fields).
Does your webmap contains a Map Tour layer? is it working fine? (can see data when zooming/panning around, can open the attribute table).
Was your new basemap using a different projection? That is supported but one error has already been reported when switching to a different projection.
Have you tried to create a Map Tour from this web map a second time?
Please share the webmap/app id and I will take a look.