You can do it with the "erase tile" action. You specify the tile to erase by it's column and row on the tilemap. If you want to delete at tile at an x and y position, the tilemap object has expressions to convert from an x, y to column and row.
To drag and drop I think you'll have to create a sprite at the tile you clicked on, erase the tile, then drag the sprite around. When you drop, you'd set the tile the sprite was over, then destroy the sprite. I don't have time currently, but it's an idea you could play around with.