Sinn Féin has accused the DUP of "holding the public to ransom for their Brexit mess".
It comes after DUP Leader Sir Jeffrey Donaldson confirmed his party will not nominate a speaker or ministers when the new Assembly meets on Friday, as part of its opposition to the Northern Ireland Protocol.
Without a Speaker, the Assembly will be prevented from carrying out business.
Sinn Féin Vice President Michelle O'Neill tweeted: "Today is the day we should be forming an Executive to put money in peoples pockets and to start to fix our health service.
"The DUP have confirmed they will punish the public and not turn up. They are disgracefully holding the public to ransom for their Brexit mess. Shameful."
Sorcha Eastwood of the Alliance Party said the DUP are "punishing us all".
"Six years ago, the DUP backed the wrong horse," she tweeted. "They gambled on a monumentally significant referendum that gave us Brexit.
"NI voted to remain, but now, six years later, they are punishing us all because they have buyer's remorse. Shameful. DUP care not one jot about NI or its people."
SDLP Leader Colum Eastwood said it is "disgraceful".
He added: "The DUP have no mandate for this as they told everyone during the election debates that the Assembly would continue."
Taoiseach Micheal Martin has said that it is "unsatisfactory" and disappointing that the DUP will block the election of an Assembly speaker.
"The people elected an Assembly, the Assembly should meet, and then the Assembly should form an executive," Mr Martin said.
"Yes there are issues that unionism has raised with us in respect of the protocol, but those issues should not prevent the establishment and convening of the Assembly and the formation of the executive."