What will happen if my product count increases to more than my plan limit?

Nothing will break! Your visitors will be able to search on your products, but they can search only on products which are within your plan limit. Let's say you are on the Starter Plan, which allows search for 1000 products. If your store has more than 1000 products at any point of time, the visitors will be able to search only within the first 1000 products.