People here on this forum have reported getting 2560x1440 at 60Hz using HDMI on the 2014 Mini. I use a 1920x1080 with a HDMI monitor at 60Hz on my 2014 Mini with no problem. Any higher resolution will be iffy in terms of the refresh rate (the official Intel max figure is 3280x2000 at 60Hz, but Apple may not adhere to that). What's the resolution you're trying to achieve? If you're trying 2560x1440 or less, what monitor are you using? The base 2014 Mini had the HD5000 GPU, the others had the Iris 5100 but I don't see any difference in terms of the max HDMI resolution/refresh capabilities.
Maybe your cable? Have you tested it with other computer/monitors?