It depends. It's one of those things where we used different ways to refer to areas . For example if we were referring to Hawaii, we may say "I am going to visit the Big Island" depending on the area. Someone could also say "I am driving through the desert", "I am driving through Sahara Desert", "I am driving through the Sahara Desert" "I am driving through the Sahara" and people would understand just fine. Also just to note, Technically Long Island , (if you are) referring to in the United States, is not legally considered an island. At least to our government, they consider it a peninsula. Random piece of informaiton when we refer to it we refer to it as an area and would say "I am moving to Long Island" the same way we would say "I am moving to Los Angeles".

Long Island is the name of the island. The Sahara is the name of the desert. I know there are other examples where we use "the" and I don't know why we do it sometimes and not other times.

In your example, “Sahara” is used as an adjective modifying desert, so you need the article “the”. Long Island is two-word noun, so you don’t use the article. Island groups that contain an adjective would need “the”: the Canary Islands, the Hawaiian Islands.

It's tricky! If it's in the name we use it. Here is a good list: 'Use 23' https://www.englishpage.com/articles/advanced-articles.htm