Not much. Without metal, you unwind most of the tech gains from World War II and later in all fields.
Electricity. None of the electronics can be done with wood. Power generation and transmission is all metal.
Transportation: The Soviets did work out how to build a stealth aircraft using wood, competing with USA’s stealth bomber; it could fly high, but it couldn’t turn fast because of sheer forces. Automobile bodies can be done with wood (although those are mostly plastic today, not metal). Ships do well with wood… except not if you want them to take the role of airplane carriers or transporting nuclear power engines. There’s scale problems with wooden ships. Submarines can be built with wood, but to very limited depth. You are not going to have meaningful space travel (rockets) without metal.
Buildings. You can’t build the dense city cores with wood — it just can’t go as tall as metal without a lot more structural reinforcement.
Medical. Modern medicine requires metal. Aside from the electronics, you have hypodermic needles (steel), splints/pins, dental implants, scalpels and other tools… the precision of metal and it’s ability to be reproduced exactly in bulk matches the demands of medicine for reproducible conditions.
Pipes. Water, oil, sewage… metal rules here. Wood gets torn up really fast in a sewage environment. Stone can build the structures, but it is just as non-renewable as metal.
Summary. Your society loses the mass agriculture, the medical advances, and the city infrastructure that make modern life possible. You lose the future of space travel without metal. You’ll lose the education infrastructure that the Internet has brought us. Focus on 19th century living for your stories.