If you want to hike levadas or visit the north and west coasts, yes. Funchal-only stays don't need one — the city has good buses, taxis are cheap, and most day trips can be done by tour or Bolt. A car becomes essential from day three for most itineraries.
Funchal itself is walkable end-to-end, and Horários do Funchal runs frequent buses to Câmara de Lobos, Cabo Girão and Ribeira Brava. Bolt operates across greater Funchal at roughly half the price of a metered taxi.
Once you want to reach trailheads (Fanal, Caldeirão Verde, Pico do Areeiro), drive the ER101 around the north coast, or visit Porto Moniz, São Vicente and Calheta, a rental is by far the easiest option. Public buses serve these places but typically once or twice a day, which kills a hiking plan.
Roads are well-maintained but steep, narrow in old villages, and the VR1/VE1 expressways have lots of short tunnels. Diesel/petrol prices are EU-mainland normal. Parking in Funchal is paid Mon–Sat 09:00–19:00.