Can an AND gate be used to toggle an HDMI connection?
Absolutely not.

First: HDMI uses TMDS signaling. This is not compatible with TTL logic gates; it's a current-mode differential protocol, very different from the single-ended outputs used in TTL logic.

Second: The TMDS signals used by HDMI run at over 1 GHz. This is far faster than the 74LS parts you're looking at can handle; indeed, I don't think it's compatible with through-hole parts at all. (The pins alone will have too much inductance for this fast of a signal.)

Consider using a dedicated HDMI switching part, such as the TI TMDS261B ("2 to 1 HDMI switch") or the Fairchild FDHDMI08 ("Wide-Bandwidth Differential Signaling HDMI Switch").