The first question you really need to ask yourself is what are your running goals? Because if your goal involves a faster pace, a faster racing performance, and an increased race time, then the weight of your shoes might make a difference. However, if your goals are more focused on running for the enjoyment, running to make friends and form community, running to relieve stress, or even running just to shed a few pounds, then the weight of your shoes is not as critical of a factor to your running (especially not as much as which shoe you choose to run in and how the shoes feel on your feet).
Obviously, for the most part, a lighter running shoe will be easier to pick up during your running stride. Lighter, more minimalist shoes are specifically designed for racing on fast, "clean", paved roads. However, they also tend to get worn down more, and lose cushioning and support faster than a bulkier shoe. In fact, overall, a lighter shoe will likely have less cushion and less support than heavier shoes. For a lot of runners, this is not really a problem if they cycle their shoes - meaning, they go back and forth between two or three pairs during their training. They will train in both lighter shoes and heavier shoes, but spend the majority of training in heavier shoes. This results both in not wearing down the minimalist shoes as much but also in getting the legs used to running in heavier shoes so that on race day, the lighter shoes just feel that much lighter. (If you tend to over pronate though, it might be harder to find a lighter shoe that offers you the ankle support and stability you need. The same goes for people with fallen arches and flatter feet, who need a more supportive arch.)
We have a great list of minimalist shoes found here, and you can read more about lightweight shoes in our article here. Overall, the time and effort you put into training in the days and weeks leading up to your race are going to matter substantially more than the weight of your shoe. The same is true for your nutrition and hydration, sleep, and recovery between workouts. Still, buying the right shoe for you is one of those easy things you can do for yourself to make race day just a bit more successful.