Why do we say תיקו Teiku when there is no answer in the Gemara?
It is important (and extra-important when teaching Gemara to beginners) to understand the basics first. The word "teiku" simply means let it stand.
Sometimes there may be questions that we just acknowledge as good questions, and we live with them for now and move on. (As some good, believing, God-fearing Jews do with this one or this one.) We are human and may not have all the answers all the time.
The Aruch (Hasholeim) Aruch tof koof, says that the word teiku means a pouch. Meaning the answer is hidden as something that is in a pouch. He then quotes Rav Yakov Saspordas that the word TEIKU could reference a contraction of תשבי יתרץ קושיות ובעיות. A similar thought is mentioned in the Tosfos Yom tov in the last mishna in Edios.
It is said to stand for תשבי יתרץ קושיות ובעיות that is תשבי who is Eliyahu Hanavi will answer the questions. This was established in Parshas Pinchas as when Zimri brought Kozbi to Moshe in front of Beis din and a large portion of Klal Yisroel and asked him is she Muttar to me? Moshe said no.He answered back how is your wife also a daughter of Yisro also a gentile allowed to you (The Ibn Ezra says this is actually the reason Moshe's children never became Kohanim?!)? Moshe had nothing to answer neither did the Beis Din or any of Klal Yisroel it was the first תיקו. One person had the answer it was Pinchas who is Eliyahu therfore we say תיקו which is an acronym for תשבי יתרץ קושיות ובעיות cause Pinchas who is Eliyahu the תשבי is the one who had the answer.'Ha'Bo'el es Ha'Kutis, Kana'in Pog'in Bo'
The Zohar, Ra'aya M'hemnah, in Parshas Tzav says that Teiku means it will always stand as a question, as it stands for Tikun minus the nun sha'arei Binah (the 50 [=Nun] gates of understanding). This is used as a signal that this question comes from the klipos (shells) and could not have an answer because the halacha it is addressing has an element of gezeirah (divine decree) to it and not fully understandable by Man.
It continues on to say that Eliyahu will answer other questions that do not end up as "Teiku". This contradicts the tradition that YS quotes. As is usually the case with sod (the hidden parts of Torah), this Zohar needs a Rebbi to explain it.