Are there any types of questions that Eliyahu (Elijah) cannot answer?

The Gemarrah often lists disputes that end in "Teiku". I understood that this term was an acronym that meant that Eliyahu will eventually answer such questions and problems. Are there any types of questions mentioned in the Gemarrah or elsewhere that Eliyahu cannot answer? (Please don't say "this one", LOL :-)

Answer

Eliyahu HaNavi is found along the chain of the mesorah from Moshe Rabenu and received the entire tradition (Introduction To Mishne Torah) and as such would be capable of answering any Halachic question. (At this point in history there were no disagreement among scholars - בראשונה לא הייתה מחלוקת בישראל אלא על הסמיכה בלבד‏ -ירושלמי, חגיגה פרק ב דף עז.) It is implied by Mishna Eduyos (8:7 and commentaries) that Eliyahu will answer Halachic questions at the end of days and resolve all scholarly disputes (Raavad ibid and Shoshanim LeDovid (Fardo)).

For non-Halachic matters, we find that no Navi (Eliyahu included) truly understood the nature of the reward in Olam Habah - only G-d knows it (Sandehrin 99a):

אמר רבי חייא בר אבא א"ר יוחנן כל הנביאים כולן לא נתנבאו אלא לימות המשיח אבל לעולם הבא (ישעיהו סד, ג) עין לא ראתה א---ם זולתך יעשה למחכה לו

We now discuss some cases where Eliyahu HaNavi either would not be followed if he would issue a ruling on a matter or he would not act on some matter.

It is tacitly implied that Eliyahu's authority is generally binding in Halacha since the Talmud writes that his opinion is followed except where it contradicts accepted Halachic practice

ר' בא רב יהודה בשם רב אם יבא אליהו ויאמר שחולצין במנעל שומעין שלו. שאין חולצין בסנדל אין שומעין לו. שהרי הרבים נהגו לחלוץ בסנדל

(Bavli Menacos 32a, Yerushami, Yevamos 12,1)

The Mishna (Eduyos ח,ז) writes that Eliyahu will not not distance families that have already been accepted into the Jewish community. The Mishna does not say that he does not personally know those families. rather, he won't act on that knowledge.

אמר רבי יהושוע, מקובל אני מרבן יוחנן בן זכאי ששמע מרבו ורבו מרבו, הלכה למשה מסיניי, שאין אלייהו בא לטמא ולטהר, לרחק ולקרב, אלא לרחק את המקורבין בזרוע, ולקרב את המרוחקין בזרוע.

A common misconception is that תיקו literally means "תשבי יתרץ קושיות ובעיות" - Eliyahu Ha'Navi will answer questions and queries. That is not correct. It literally means the question will stand unanswered. "תשבי יתרץ קושיות ובעיות" is a derush and is found in early sources ( Shla Torah She bal Peh, Tosfis Yom Tov Eduyos ibid)

For a discussion as to whether Eliyahu's authority is based on Nevuah or Chachmah see Sdei Chemed Ma'arechet Mem, Klal Lamed Chet, sh"ut Chsssam Sofer 6:98 and introduction to Sh"ut Min Hashamayim R Margolios edition

For an extensive discussion on this and related issues see Encyclopedia Talmudit entry on Eliyahu.