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On OpenStreetMap, some objects are overground (tagged with layer=1, layer=2, layer=3 and so on), and some are underground (tagged with layer=-1, layer=-2, etc.).

Browsing the base OSM website (https://www.openstreetmap.org/), I didn't find a way to only display overground (or underground objects).

How would it be possible to do that?

More generally, how would it be possible to keep the OSM base style but to render only some features of the map (filtering them by tag for instance)? I have used Overpass Turbo, but it only highlights the selected features, without hiding the non-selected ones.

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OpenLevelUp! is a service that renders layer and level tags on OpenStreetMap Carto.

If you want to fully render in the style of OSM Carto, you would have to export the Overpass Turbo elements as a .osm file and render it on your own tile server, instructions available here.

Alternatively, if you aren't comfortable with setting up PostgreSQL, you can import the .osm file into Maperitive with an OSM Carto stylesheet, though this probably won't look exactly the same as actual rendering.

As til_b mentioned, the osm.org tileserver uses raster tiles, so these cannot be modified by the client.

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On openstreetmap.org you cannot change the rendering.

The way to go is using vector tiles, https://wiki.openstreetmap.org/wiki/Vector_tiles .

I cannot find a solution for exactly your usecase ("keep the openstreetmap.org rendering, and then add some specifics").

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