All GPUs support any refresh rate. As for the ports, you need DisplayPort 1.2 or better. 1080p 240Hz is supported. And mind you, DP 1.2 is very old, and DP 1.4 (it is a 2016 standard) is definitely supported by the RTX 2080.
In fact, its DP supports up to 1440p @240Hz, 4K @144Hz, 5K @60Hz or even 8K @30Hz.
With DSC and 4:2:0, DP can even do 8K @144Hz!
You should not be worrying about the card supporting 240Hz, but rather, will the games you play maintain 240+ FPS constantly on the settings you play on.