England and Wales Cricket Board (ECB) rules need to be altered to remove the right of all 18 counties to play in the new tournament.
Essex follow Middlesex in their refusal to support the change.
But Durham chairman Sir Ian Botham said his county were fully committed in their support for the new tournament.
He said: "It's financially very sensible to do it and if we prepare properly and come up with the right formula throughout the summer - just look at the success of the Big Bash in Australia and the IPL - then it's proven that the format has great benefit."
Meanwhile, Essex chairman John Faragher said the County Championship, one-day and T20 competitions "must be protected".
The changes require the support of 31 out of 41 of the ECB's voting members.
Faragher continued: "We are focused on expanding cricket in Essex, East Anglia and Metropolitan London, ensuring there are opportunities for all age and ability groups, male and female, to be actively involved in the game.
"We believe that as a result of the proposed changes, these opportunities will be reduced, that our income overall will suffer and the first-class game will be diminished, in contradiction to the ECB's objective which is to grow the game in this country - an objective that is unlikely to be advanced by a competition which would exclude large areas of the country from any involvement in it."
The 41 voting members comprise the 18 first-class counties, 21 recreational boards, the Minor Counties Cricket Association, and MCC - owners of Lord's and therefore Middlesex's landlord.
A number of counties have come out in support of the proposals for the new tournament, with Derbyshire, Leicestershire, Somerset, Sussex and Yorkshire all announcing that they will back the ECB rule change, while others including Glamorgan, Hampshire and Warwickshire have been vocal in their support for a city-based competition.
Kent have asked their members and supporters to give them further feedback before making a decision, while Surrey director of cricket Alec Stewart has expressed concern over the details of the new event.

Hilda Marcel died in 1987 from lung failure and her husband Eugenios died from prostate cancer seven years later.
Their mummified bodies were kept in a basement flat in the city's Gilmore Place, which was regularly visited by a son of the couple, until 2002.
Since their discovery, the bodies have been stored in the city's mortuary.
Edinburgh city council has been trying to get the family's consent to bury Mr and Mrs Marcel, but this has been repeatedly refused.
Now, almost 30 years after Mrs Marcel died, the council has lodged a petition at the Court of Session to seek permission to dispose of their bodies.
Prof Roderick Paisley from Aberdeen University's Law School said: "The wheels of justice do grind slowly, and they appear to be grinding incredibly slowly on this particular hearse.
"I think this should have been dealt with three or four years ago.
"I do appreciate there are sensitivities here, but the law is reasonably clear."
A police investigation was carried out when the bodies were found in 2002, but no charges were brought.
Death certificates had been correctly issued and the pair had died of natural causes.
Cameron Rose, leader of the Conservative group on Edinburgh City Council, said it was an "extraordinary story".
He added that there needed to be "sensitivity for the families".
The Court of Session has now agreed to hear the case, although so far no date has been set.
Essex have become the second county to announce their intention to stand against a new city-based Twenty20 competition, planned for 2020.
[NEXT_CONCEPT]
A council has launched a legal action in an attempt to bury the mummified bodies of a husband and wife who died decades ago in Edinburgh.