Mac mini Multiple Monitors

Officially the 2014 Mac Mini only supports 2 displays. One on the HDMI port and one on the a Mini DisplayPort.

see > https://support.apple.com/kb/sp710?locale=en_US

I'm not well versed with the Apple TVs, but it looks like both the HD and new 4K boxes require using the Mac Mini's HDMI port.

see > https://www.apple.com/tv/

However you may be able to get three monitors to work, if you connect one to the HDMI Port, one to the Mini DisplayPort and a third to a USB 3.0 port using a USB 3.0 to Video Display adapter.

see > https://discussions.apple.com/thread/6682888

then > https://itectec.com/askdifferent/mac-hook-up-3-monitors-to-mac-mini-late-2014/

and > https://www.tripplite.com/usb-3-0-to-dvi-hdmi-dual-monitor-video-display-adapter-microsoft-surface~U344001HDDVI