Salvo's in Headingley said it had to lay off nine staff members, one of whom was full-time, due to payments under the job retention scheme having yet to be paid out.
Meanwhile, staff who lost their jobs say they had been left high and dry, with some unable to apply for financial support.
Owner John Dammone said Salvo's had managed to furlough 22 employees, but that without yet receiving payment from the Government, it had "no alternative" but to lay off staff in order to survive the lockdown.
He said: "Whilst the government has announced a furlough scheme, the reality is that no payments have been made under the scheme.
"Accordingly the current position is that the cost of the scheme (80 per cent of salaries) has to be funded and paid on a weekly basis by the employer. The government has not made any payments whatsoever to employers for furloughed staff.
"So, whilst having no income coming into the business, we are totally reliant on the limited additional banking facilities my family have raised. We have made an application for Government support through the furlough scheme once the HMRC portal opened but as you will be aware monies have not been released yet."
Mr Dammone currently runs the business after his father, Salvatore Dammone, opened the Otley Road restaurant in 1976.
He added: "If we furloughed all our staff we would likely run out of money before the government furlough scheme paid out to us and therefore jeopardise the ability of the restaurant to remain in business at all.
"It remains a source of genuine regret that we have been forced to let valued members of the team go but we simply have no alternative if the business is to have any hope of surviving and reopening and providing future employment to the rest of the team."
One staff member, who asked to remain anonymous, said: "It has been weeks now since we were formally told we had lost our jobs and were sent P45s. We have sent a letter and two follow up emails on behalf of 20 of us asking for more clarity.
"Another of my colleagues was never even properly told he had been laid off. He only found out the Friday after the lockdown was announced when he was sent his P45, so was made to assume he had lost his job.
"All we are wanting from this is to be placed on furlough. I am lucky that I have savings but some of the staff have had to turn to their families for support. One of my colleagues has had to return to Poland where his family live."
TUC Regional Secretary Bill Adams said: “Employers who take Government money to stay afloat must keep their own staff afloat by furloughing them. They can’t expect a free bail-out, when their workers will be struggling to put food on the table.
“The government’s job retention scheme was designed to help companies keep their staff on, to limit the economic crisis. It is immoral and bad for the economy to cut hardworking employees adrift during a global pandemic.
“Employers have no excuse not to furlough workers, with the scheme now open and back-dated pay available.
“I am calling on local businesses to do the right thing and keep staff on. Unions will not hesitate to call out bad employers.”
A spokesman for HMRC said: “The Coronavirus Job Retention Scheme launched on 20 April and wages for furloughed employees have already begun to be paid, just six working days after applying.”
“The scheme supports UK employers to keep employees on the payroll, even if they can’t afford to pay them, so employees still get at least 80 per cent of their regular pay up to a maximum of £2,500.
“We do not comment on identifiable taxpayers.”