Does my 1965 GTO Convertible have to be running to raise and lower the top.

Is the battery weak? Try turning the key to the running position. Don't force it.

With a strong battery you should be able to operate it without the engine running. I believe you can open it manually but it should be done slowly and carefully and you will need a helper so it can be lifted up and pushed back evenly to avoid bending things. I can do that with mine (although it is 30 years newer and they're not all the same, so be careful). Specialty shops like convertibleparts.com will be able to give you some guidance as well if you need it.

Put it a fresh battery or use a quick starter to provide power and it should work from the dash switch.