How to close all tooltips defined against the map object?

You can get all layers from your map, and remove them from the map. To avoid removing the tile layer (background), filter with if(layer.options.pane === "tooltipPane").

map.eachLayer(function(layer) {
    if(layer.options.pane === "tooltipPane") layer.removeFrom(map);
});

Fiddle: https://jsfiddle.net/3v7hd2vx/392/