I have been playing kitten game and I have noticed that if I leave the game for a year the game runs fine and no kittens starve. Is there a consequence to this or can I just leave my game running to get resources?
Depends on how far into the game you are. If your catnip production is high enough such that a -90% cold winter comes along and your kittens don't starve, than there are no negative consequences. But, it's possible you'll miss out on some advantages - based on how far you are into the game.
At the very beginning of the game (before Astronomical events) you won't 'miss out' on anything, but your resource limits aren't likely to be high enough to make this a useful strategy.
Once Astronomical Events and Hunters are unlocked, you'll have to pay more attention to the game if you want to make sure you aren't maxing out your catpower (thereby losing valuable ivory and furs, which equals more production) or missing Astronomical events - which give you science and a starchart.
At the end of the game, once you have a Mint (or a few) (which produces furs and ivory) and have the upgrade that automatically registers Astronomical Events, the only thing you have to watch out for is maxing out your resource pools.
TLDR; The only things you have to worry about are negative producing resources, click required resources (Astronomical Events and Hunting), and maxing out any resource pools.
There's a few things you'll be missing. They are events that you need to click (Astronomical events) that you'll miss the science from them.
There might be other events like those but I'm not that far out to know but I suspect there might be.
You won't be able to send hunters to hunt and gain the happiness bonus from the resources they get as that need a click to do.
If you are sure your catnip reserve will never get to 0 even with all seasons being cold ones, it should be safe kitten-wise.