Can I output 1080p at 240hz from my 2017 MacBook Pro 13inch with Touch Bar

Solution 1:

Unfortunately, no.

Because your MacBook only has USB-C Thunderbolt, you will need a USB-C Thunderbolt to DisplayPort adapter. This Apple USB-C to DisplayPort adapter states that it can provide up to 60hz. This Apple article covers connecting UHD monitors with your Mac and states (paraphrased):

The MacBook Pro (2016 and later) can support these resolutions and refresh rates over HDMI 1.4b when using the USB-C Digital AV Multiport Adapter with macOS Sierra 10.12 or later: Up to 1080p at 60Hz 3840 x 2160 at 30Hz

Allan is correct, you will probably want an active Thunderbolt to Displayport adapter. I forget USB-C and Thunderbolt are not interchangeable terms.

Solution 2:

Hardware

MacBook: 2017 15" MacBook Pro (maxed out)
Display: AOC AGON AG251FWG2 (1920x1080@240Hz)

Using DisplayPort

I used this generic 4K@60Hz USB-C to DP adapter and it correctly handled 1080p@240Hz resolution on my MacBook.

Using HDMI

On my MacBook, I only get 1080p@120Hz since I only have a 4K@30Hz adapter (which would translate to 1080p@120Hz, logically).

However, on my Hackintosh (Vega 64), the maximum refresh rate I get is 1080p@144Hz. So there’s probably also a macOS + HDMI limitation.

FreeSync

Neither on my MacBook Pro, nor on my Hackintosh, FreeSync works in any of these configurations. I think it’s not supported at all by the macOS drivers.