The answer to this question really depends on what you mean by “support” and “video”:
If you “just” want to query/receive/transfer DICOM videos, Orthanc will work fine either with 2D+t or real videos (because Orthanc is a Vendor Neutral Archive). This distinction is also discussed in another FAQ entry.
It is easy to extract and download the raw video embedded in the DICOM instance using the REST API of Orthanc.
If you also want to play the videos, the Osimis Web Viewer plugin is able to play H.264 (MPEG4) videos and 2D+t (cine) sequences but not MPEG2 videos that currently can not be played by Web browsers.
If your video is a 2D+t (cine) sequence, Orthanc can also display it inside a Web browser by at least 2 different means:
To summarize, if your video is not encoded with MPEG2, OR if you do not need to play the video within a Web browser, Orthanc actually supports video if the Osimis Web viewer plugin is installed.