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I have ZSNES 1.5 and I added a game genie code, but I'm not sure how to turn it on/off.

Is it automatic? I added a code for infinite lives in Super Double Dragon, but it doesn't appear to be working.

when you add the game genie code it is automaticly turned on, but in some games it will not work untell the game is restarted. as for turning on/off the game genie codes: goto the cheat screen as shown below

where it says "on" in the cheat display, just double click the word "on" to turn the cheat on or off.
i suggest that you enter your codes during the intro of a game and not during the actual game play.