On most of its systems Apple does not support HDMI output, and in order to use HDMI people have had to use adapters such as Apple's DisplayPort-to-HDMI adapter. For the most part these work fine for sending audio and video information to various devices, but some people have found after upgrading to OS X Lion that their HDMI connections are either not showing video, not playing audio, or both.
Unfortunately the root cause of this problem is not too apparent, as the HDMI behaviors seem to be different for different setups. In some instances it happens intermittently, and in others only one aspect of the HDMI connection is working (either video or audio). It may be that the problem is rooted in video drivers and could be fixed by a simple update from Apple, but so far no updates for Lion have addressed the problem for those who have experienced it.
If you are noticing odd HDMI connectivity issues after updating to Lion, here are some suggestions which may help remedy the situation:
Reset PRAM and SMC
The PRAM and SMC hold hardware variables that may affect audio and video output of the system, especially if there is any corruption in these settings. See these links to reset the PRAM and SMC on your system.
Try a new HDMI adapter
The HDMI adapter you are using may be contributing to the problem, so one option is to try a new one. Some people have had success by going with a relatively cheap adapter. Apple has a knowledgebase article suggesting a couple of HDMI adaptors that should work properly, including those from Moshi and Griffin.
Reconfigure audio formats on the system
For those who are missing audio in the HDMI connection, you can try using Apple's Audio MIDI setup utility to change the audio format used on the system. Select the output channel, then configure your speakers to be different formats such as 5.1 surround or 7.1 surround, and then change the format for these channels to be a different sampling rate (such as 48KHz as opposed to 44.1KHz), and have it be 24-bit instead of 16-bit, or vice versa. Different combinations of these settings may yield different results, depending on your system's setup.
Change the number of displays
If you have multiple displays on your system (including an internal display in addition to the use of the HDMI adapter), then try turning some of them off or disabling them. For laptop systems you can try closing the display to put it in "clamshell" mode and disable output to the internal display so only the HDMI adapter is used.