how flexible do you have to be for diving?

If you want to be a _good_ diver, then yes you do need to be flexible.

If you can't touch your toes without bending your knees, you won't be able to e.g. do a tight 'pike' position -- and a loose pike will not rotate as fast. Piked dives have a higher 'degree of difficulty' (i.e. score higher) than tucked dives, but if you can't do a good pike, you might have to stick to doing your somersaults tucked in order to be able to make a vertical entry.

Also, since diving requires you to change body positions very quickly, being flexible means that you are less likely to overstretch/injure your muscles/tendons.

You do not have to be. I know a girl who went to state for high school diving and could not even touch her toes. It doesn't take much talent or flexibility.

You don't have to be flexible to dive, but it will be a lot easier if you know how to do flips or do gymnastics. You can still learn how to be a good diver even if you aren't flexible. Good luck :)