Do police need a warrant to place a GPS tracker on your car?
Yes — the Supreme Court in United States v. Jones held that physically attaching a GPS device to a vehicle is a Fourth Amendment search, so law enforcement generally needs a warrant to do so.
Is it illegal to remove or throw away a police GPS tracker you find on your car?
In many cases removing an unmarked tracker is not theft or tampering; courts (including an Indiana supreme court decision) have rejected theft claims when removal doesn't show intent to permanently deprive the owner and circumstances suggest abandonment by police. State law can vary.
Can police charge you with theft to justify searching your home after you remove a tracker?
Police have attempted that tactic (as in Derek Hearing's case) to obtain a search warrant, but higher courts have sometimes invalidated those warrants when removal didn't provide probable cause for theft.
What should you do if you find a GPS tracker on your vehicle?
Document the device (photos/video), consider using a GPS-tracker detector to check for more units, contact an attorney promptly, and be cautious about destroying evidence if a criminal investigation might be involved.