When does a tooth need extraction?

Unrestorability: When a tooth is so broken down that no amount of restorative dentistry would produce a predictable and long lasting result, the tooth would be deemed hopeless. If periodontally involved, then the tooth would be extracted if surgery could not help the condition. Nowadays it is often better to replace teeth with implants than to try to restore a tooth with a poor prognosis.

Variety of reasons: Impacted wisdom teeth, supernumerary teeth, broken or damaged by decay, braces treatment, any infections, tumors or bone disease, etc.

Depends: depends on restorablity and its supporting structures (tissue and bone). It also depends on predictability (ie if you will only get 50% or less predictability out of the future treatment most likely an implant at 95-99% success is a better investment). Sadly it also comes down the patients ability to pay for predictable treatments (root canals) etc, I deal with medicaid and many will not do RCTs

Extraction: A tooth needs to be removed when keeping it will cause more harm than removing it. Many reasons. Non-restorable damage, infection, fracture, malposed, impacted, extra tooth, loss of bone/gum support, just to name a few reasons.