In the sefer Noam Elimelech, many of his Torah pieces have shorter pieces within them that have a few asterisks before and after the piece. What is the significance of these markings?
I'm not sure what you're referring to specifically; are you talking about dots between paragraphs? That may be just delineating separate pieces.
On the other hand, in the earliest editions of the text there are asterisks embedded within the text - those follow seemingly fragmented statements and comments. According to Rav Tal Moshe Zwecker:
In the original printing of Noam Elimelech there were stars or asterisks found throughout the passages. Tradition has it that these stars have some meaning. In the sefer Devarim Areivim,25 it is written: “In the first edition of the sefer, there are in many places small stars which allude to some secret meaning.” I also heard in the name of the Klausenberger that in the heavens above (possibly meaning the stars themselves) there are thousands of “commentaries” on the small stars found in the work Noam Elimelech. I have therefore included those stars wherever they appear in the original text.
(His translation in free PDF can be found here ).
The Noam Elimelech is a deep, mystical text on the Torah; in Hasidic lore, it is one of three essential books, along with the Tanya and Likkutei MoHaran. They are referred to respectively as the Sefer Tzaddikim the Sefer Beinonim and the Sefer Reshaim (unattributed quote; although I heard it from Rav Moshe Weinberger and others).