The pixel resolution is supported just fine. There's no limitation on the size of display supported, since the display device is doing all the work of actually powering the panel - you are just sending signal information to it for the TV to process out to whatever panel size it has. A 19" 1080p native resolution display has the same number of pixels as a 55" 1080p native resolution display.
Your main concern with the older mini is the processing capability in regards to the bitrate or framerate of the video. A Core 2 Duo 1.83 ghz does well handling most HD video signals from online streaming sources as I can attest. If your mini is definitely a 2006 Core Duo 1.83, it may struggle a little more, but will probably be just fine. Most decent video sources available online are adaptive, and will detect what quality and bitrate your system can support, as well as what your internet connection speed can support. If your connection is too slow, it may present a greater limitation on video quality than your hardware's capabilities.
Video from the hard drive, like streaming content, depends on the bitrate used when the video was encoded. CDs and DVDs will be just fine.