Your challenge will be largely moving all that data. You will need to specify the cables and driving and receiving electronics for extremely high frequency operation. Consider:
$$1920 \times 1080 = 2073600 \text{ pixels}$$
If you want RGB color, with 8 bits per color, then:
$$ 2073600 \text{ pixels} \times 3 \text{ colors} \times 8 \text{ bits} =
49766400 \text{ bits per frame} $$
Then, 240 frames per second:
$$ 49766400 \times 240 = 11943936000 \text{ bits per second} \approx 12Gb/s $$
The HDMI 2.0 specification (according to Wikipedia) can transport up to 14.4Gb/s, so at 1080p resolution, your 240Hz could just fit. Interestingly, Wikipedia also says the maximum resolution supported is 3840×2160p60, which if you do the same math as above, works out to exactly the same data rate we just calculated.
I don't know in detail the HDMI or DVI specifications, but I'm not aware of any hard limit which would prevent you from signaling at a 240Hz frame rate. Certainly 120Hz devices are commonly available. I suspect your challenges will be in finding video sources and sinks that support that framerate, and finding HDMI interfaces and cables that support HDMI 2.0, simply because it is so new.