Unfortunately, since your extension hair is not growing out of your scalp, it doesn't really get the benefit of your scalp oils as your natural hair does.
Dry extension hair can be especially annoying in the winter and in places where the air is extremely dry.
So, just like your skin, your hair needs the moisture. Sometimes many products (especially cheap store bought ones) can work against this so be sure you check the ingredients and only use products without drying sulfates and alcohols.
You can use some oils (not on your bonds!) just on the ends of your hair but sure you do not use a hot iron on the extension hair after oil has been put on it. You aren't frying an egg and we don't want you to cook your extension hair either. It is best to style your hair and then add only a small amount of oil for moisture and shine. But, then don't forget and run a hot iron on it before washing the oils out!
Always, product-free extension hair is better so washing your extensions less and using a looper brush to brush your natural scalp oils through to the ends of even your extension hair can help, too.