While it’s possible to export height from Tel Aviv’s GIS, importing it into OSM is not going to be easy, because you’d need some way to match every building in Tel Aviv’s GIS to the buildings in OSM, and there’s no one-to-one mapping of these. The buildings that do have height data are ones where someone who cared about them decided to contribute the height data, either by specifying the number of floors, or the actual building height.
In the website you linked to, all buildings are 3D, but those that don’t have height data are all displayed with the same fake height. The shadows it shows are not “real”, because the height of the buildings displayed there is not real.
OSM doesn’t have tiered access to data, if you donate to OSM you don’t get more data than you can download yourself.
I suggest that if you want to enter height data, do it manually for buildings you care about.
