Unplug the lamp and allow it to cool.
Rub the bronze part of the lamp with a dry, soft cloth to remove some of the most obvious tarnish.
Mix warm water and a few squirts of dishwashing liquid in a bowl. Polish the brass with this solution, which gently removes a bit more of the tarnish. Dry the lamp with a soft towel afterward.
Drip mineral spirits onto a dry cloth and carefully polish the lamp. Mineral spirits are a solvent and slightly harsher than the other methods so take care not to rub too hard.
Use an abrasive cream for bronze, if there is still too much tarnish. Polish the tarnished parts of the lamp gently, increasing the pressure as necessary. Wipe off remaining traces of the cream with a damp cloth and dry the lamp with the towel.