Lawyers for the civil rights organisation argued on Friday that its legal challenge, brought to ensure Mr Johnson "acts within the law", should be heard immediately.
But their application for an urgent hearing was rejected by Court of Appeal judges.
Liberty asked for an immediate hearing of its case against Mr Johnson, which it is bringing to make sure he complies with the European Union (Withdrawal) (No 2) Act, commonly referred to as the Benn Act.
The Act requires the PM to seek an extension to the current Brexit deadline of October 31 if no deal is reached with the EU, and approved by Parliament, by Saturday.
The organisation's application for an expedited hearing was previously rejected by Mr Justice Supperstone, a High Court judge, following consideration of the court papers.
Richard Hermer QC, for Liberty, said: "It is at least being considered by the defendant that he is fully entitled, notwithstanding the terms of the Act, to seek to procure a refusal from the European Council."
Mr Hermer argued that any steps taken by Mr Johnson which are "intended and likely" to result in the European Council refusing to agree an extension to the Article 50 period are likely to result in "irreversible" consequences.
He added: "Accordingly, in order to avoid irremediable consequences of fundamental constitutional importance, it is essential that the lawfulness of those steps is determined before such irreversible consequences are caused.
"If the application is not considered until after October 19, it will be too late for the court to grant any effective relief."
However, the application was refused by the Lord Chief Justice Lord Burnett, Master of the Rolls Sir Terence Etherton and President of the Queen's Bench Division Dame Victoria Sharp.
Lord Burnett said: "It is clear to us that Mr Justice Supperstone made no error of principle or acted outside the proper limit of his judicial discretion in making the case management decision under challenge.
"We will give our full reasons in due course in writing."
Government lawyers said Liberty's case does not need to proceed urgently, as there is "ample time" for it to be heard before the UK's planned exit from the EU on October 31.
Sir James Eadie QC, for the Prime Minister, said: "The Prime Minister is well aware of his duty to obey the law ... and is and will continue to be advised in the usual way on any issues as to the lawfulness of his proposed actions."
No date has been set for the hearing of Liberty's High Court challenge but it could be as early as Monday.
Liberty announced it was bringing the challenge in September, saying that it was "gravely concerned" by media reports which suggested Mr Johnson intended to flout Brexit legislation.
The organisation is seeking a declaration that the Prime Minister cannot take steps which would be likely to result in the EU refusing to agree to an extension, as required by the Benn Act.
In a statement ahead of the hearing, Liberty director Martha Spurrier said: "This case is about ensuring that the Government - whoever it is, or whatever its intention - acts within the law.
She added: "These are extraordinary political times - but that doesn't mean the Government can act beyond the law.
"Any government which defies parliamentary process and law because it doesn't serve their aims should be a cause of concern for everyone."
Liberty, which is not affiliated with any political party, said it is "neutral" on Brexit and that the case is not concerned with how or whether the UK leaves the EU.
Parliament is due to vote on the deal on Saturday but Mr Johnson may struggle to get it through as the Democratic Unionist Party has voiced opposition to the agreement.
A crowdfunder to support the case has raised nearly £20,500