Summary of the opinions on today's Facebook feed:
Harav Benny Lau (Heb) - Rav Eliyahu does not accept the State's law ( including equality to all), Harav Eliyahu is not sympathetic (and worse) to those who made Aliyah but may not be halachically Jewish.
Harav Bazak (Heb) - A Chief Rabbi can't be someone who holds what even he admits are "extreme" positions - specifically regarding Israeli Arabs (gives the example of transfer). Secondly Rav Eliyahu often expresses himself badly and delegitimizes the opinions of rabbis he disagrees with.
I would have gone further on each of the points, but I think it is important that the more liberal rabbis of the Dati Leumi movement have come out clearly against Harav Eliyahu. He is not a valid candidate, nor is he representative of Religious Zionism.
HOWEVER, I'm against him being blocked from running for the position by the Attorney General. Through his own choice, the Attorney General has previously decided not to prosecute Rav Eliyahu for his outlandish comments, and even worse he allowed him to remain a tax-payer supported communal rabbi. There is no general "Not a nice guy" or "No Racists" criteria in the law for Chief Rabbi. As such the Attorney General should not take it upon himself to create criteria where none exist. (See also this letter).