VGA and HDMI are not one-way; besides the main send-pictures functionality, there is some low-bandwidth bidirectional communication. This is how a computer can “know” that a new display was connected, and what resolution to use on that display. In the case of VGA, this was a backported feature (VGA displays from the early 1990s could not do that).
Theoretically, this could be exploited, if (and only if) there is some weakness in the receiving part (the display card and its driver). The protocol is not complex (no TCP/IP stack involved) so exploitable security holes should be rare.
Usually, a malicious display can do much more evil by simply logging a copy of everything that is displayed. For instance, I have an account in a bank, such that for online banking I “type” a password by clicking on semi-randomly located buttons. A screen logger grabs the password easily in such conditions (ironically, that system was meant to thwart keyloggers). Therefore, the usual wisdom is that if your display is malicious, then you already have bigger problems.
Note: some displays also double as a USB hub — and weaknesses in USB protocols have been exploited (e.g. the “PS3 Jailbreak” from 2010). For common PC, the USB link is an extra plug, but these things may change. For instance, Apple computers now use the Thunderbolt interface which is fully bidirectional and uses a rather complex protocol, where security holes are quite plausible.