A mobile phone is typically composed of about 40% of plastic, 32% of non-ferrous metal, 20% of glass and ceramics, 3% of ferrous metal and 5% other . Metals referred to in ref  are iron, copper (16 g), silver (0.35 g), gold (0.034 g), platinum and palladium.
Ref  mentions additionally the use of aluminium, magnesium, tin, cobalt, lead, nickel, cadmium and nickel. The document provides some guidance on best practices for the end-of-life treatment of mobile phones.
Ref  refers to total 60 elements being used in mobile phones, mentioning from the metals family (in addition to above metals) tantalum, neodynium and indium. It gives good information about the recycling value-chain and its challenges.
The above three paragraphs list a total of 16 metals used in mobile phones. Based on the total use of 60 elements, and the observation that only 20-30 of the world’s 118 elements are non-metallic, the number of metallic elements used in mobile phones could be as high as 30-40.
The video “Ground Rules: Mining Right for a Sustainable Future”  mentions at minute 6 the number of 42 different metals used in a phone.
Update May 2020:
Ref  dissects a modern 5G phone giving a good indication where copper is used in a mobile phone, for wiring, heat sinks and heat transfer foils.
 http://www.basel.int/Portals/4/download.aspx?d=UNEP-CHW-COP.11-SIDE.01A-Photoexhibition.English.pdf (not dated – checked October 2018)
 http://www.basel.int/Implementation/TechnicalAssistance/Partnerships/MPPI/Overview/tabid/3268/Default.aspx (2012 – checked October 2018)
 https://www.chemistryworld.com/features/smartphone-recycling/2500497.article (2017 – checked October 2018)
 https://www.youtube.com/watch?v=CWt36I8JgVQ (2011 – checked January 2020)
 Product environmental report – iPhone Pro 11 (2019 – checked January 2020)
 https://youtu.be/TLdzV7gw0fA (2020 – checked May 2020)