Is there a function in the CryEngine Lua API to open the
I cannot find it in the Lua (
ScriptBind) API reference.
Do you mean to open the console by lua script or execute a console command from lua? As far as that goes, it would be hell of a workaround to run console commands by lua. It would be much easier to expose the c++ function that does the things you want (for example the function which the console command calls) to the lua engine and run it directly (refer to this).
Just in case you want to invoke lua from the console, simply run your script from the console by putting a # in front of it (This only works in Sandbox or when launching with -DEVMODE).