Should you leave the pilot light on your central air furnace on or off during the off season?

My A/C guy said I should turn it and the gas off all summer... I usually don't...

Gas off until you get close to needing heating.
When it gets cool and you are thinking about lighting it, that is the best time for annual maint on the inside unit. Bleach down the drain to clear away the slime that has formed over the summer, clean the blower cage and the AC coils, lube the blower motor (if it has oil ports), inspect the vent pipe and heat exchange, give the ducts a look over from leaks and kinks.

If it's your only device that uses gas you may as well turn it off. If you have other devices that use gas then turn the valve off to the unit. No sense in putting all that extra use on the thermal coupling and other parts.
Just my opinion and I'm not a professional gas guy.

If it is old enough to have a standing pilot, it is has exceeded its life expectancy.

Water is a byproduct of combustion. If the pilot is left burning, the moisture from the pilot flame, under certain conditions, can cause rust on the main burner and heat exchanger.

It is usually a combination of events that caused this theory.

IE. Fuel rich pilot flame, low cooling set point causing return air to be below dew point, furnace located in unconditioned space.

Therefore it is generally considered best to turn pilot off and deal with relighting in fall.

Drops mic. Moonwalks off stage.

Off. That small flame produces a lot of heat. Thermocouple needs to be checked and unit cleaned every fall.

Yep. Just got a new one the other day with electronic ignition...SWEET.

If it has a steady pilot light, it does not have an inducer blower with a proving switch that would prevent the main burner(s) from lighting should the flue pipe (or cap) become restricted due to bird/hornet nests. A standing pilot light will prevent said nests. Needless to say that lit main burners with a restricted/clogged flue will result in carbon monoxide within the living space (I can personally attest to this scenario). This becomes a maintenance issue: leave the pilot on and decrease the life of the thermocouple and the heat exchanger or be dang sure and vigilant to ensure an unrestricted flue /vent system. Best and safest thing to do is replace that type of old gas furnace.

Stays on... sensors around it just in case

Not sure, but a water heater has vac on the dial for a reason. Vacations!! My buddy left his on high and came back from a hunting trip to a flooded house. The metal water heater exploded because it could not release its heat pressure.