What happens when you connect two HDMI output devices

The two devices will attempt to handshake with each other. Since they are both output devices, they will not communicate. Fortunately, no damage will occur, the engineers figured people would attempt to do this. You can not change an input to an output, or an output to an input. Their direction is hardware bound, and can not be modified with software.

Would it be possible to download drivers to the laptop to actually let me use the HDMI port as video/audio input, and if so, where could I find these drivers?

It would depend on the laptop hardware, but I seriously doubt that such functionality exists in any consumer hardware. For an analogy, see USB Host Mode. USB as implemented on most consumer electronics (such as the Android link) is done in Client mode, but if the hardware supports it it can be used in a different mode. This is possible because, like HDMI (and the reason for the analogy) is that both sides of the connector could be (not necessarily are) the same physical design.