In my opinion, turning off the pilot is better for the appliance in a number of ways.
Below Article is by the “Gasman”
Here is my reasoning:
1. The thermocouple which monitors the pilot flame has a limited life expectancy, being in the flame constantly.
Turning the pilot off at the end of the heating season will prolong the life expectancy of the thermocouple or thermopile, as the case may be.

2. Contrary to the popular myth that pilots will keep a system dry, at roughly 1000 Btu’s per hour, they do not generate a lot of heat, certainly not enough to keep both the chimney and the appliance dry.
In fact, the opposite is true in many cases.
A constant pilot under cooler conditions will cause condensation to form on the inside of the firebox or the furnace heat exchanger, the condensation is acidic and will cause corrosion of any parts it comes into contact with, over time.
In the case of a gas fireplace with a glass front, the minimal amount of by-products of combustion will quickly condense on the glass under certain conditions and will leave behind that white residue that gas fireplace owners are so familiar with.
So, in addition to burning approximately 20,000 btu’s of gas per day for no good reason, you are shortening the life expectancy of the thermocouple or the thermopile, and you could be shortening the life of the appliance through corrosion.
With fireplaces, I see an increase in those hard to remove deposits, on the glass which require a special ceramic glass cleanser to remove.
My experience has been that homeowners who regularly turn off the pilot (and I recommend turning off the main gas supply to that appliance, as well) save money on gas, the appliance will last longer, look better and components such as the thermocouple will last years longer.
As for bugs that find their way into the burner compartment. It’s usually a good idea to have the appliance cleaned and checked at the beginning of each heating season.

I’ve actually seen a few cases where the pilot may have attracted critters to nest in the vent, if they can, because the pilot provided just the right amount of central heating for their new condo.