How do I get rid of a car that I own?

Solution 1:

If your car is not insured you can simply destroy it and you will be rid of it.

For insured cars it is a bit trickier:

• If your car is worth less than $50k you will be able to sell it to Los Santos Customs and get rid of it permanently that way.
  
• If the car exceeds that worth you can replace it with another car when your garage is filled completely. Just drive a car into your full garage and you will get the warning that the garage is full and if you wish to replace a car. After the loading screen you will be prompted to choose a car for replacement.
  Be careful with pressing the "OK"-button as to not accidentally lose the first car on the list, if it is not the one you want replaced.

Solution 2:

You can actually sell vehicles to the Los Santos Customs, but only one every so often one every 45 mins

Solution 3:

Pull the car that you want to remove out of your garage, then call your mechanic and ask him to bring you another vehicle from your garage. This will then destroy the vehicle, I found this out the hard way lol.