What is the smell of 'burning' metal?
According to the Canadian Centre for Occupational Health and Safety page Welding - Fumes And Gases, the welding fumes are


  a complex mixture of metallic oxides, silicates and fluorides. Fumes are formed when a metal is heated above its boiling point and its vapours condense into very fine, particles (solid particulates).


This composition varies dependent on the composition of metals being welded and fluxes.  The website lists several main types and their associated health risks.  Often associated with metallic oxides are some pungent/noticeable gases:


Gases produced from the thermal breakdown of coatings, from welding and cutting processes, stated in the article linked as being examples of "welding gases" (examples from the linked website include ozone) 
And depending on the coating that may be on the metal, organic vapours (including aldehydes, diisocyanates, phosgene and phosphine)