The remains of 30 destitute men believed to be American veterans remain unburied in the Los Angeles County coroner's office because of cutbacks three months ago in the Veterans Administration funeral allowance program.

The plight of the impoverished veterans-- officials are unsure of the precise number because fingerprint checks have not been completed--has ignited an emotional debate over the proper role of the federal government and the impact of President Reagan's budget cuts on state and local governments.

Last summer's budget reconciliation bill cut the funeral allowance for the families of most veterans by $300; only wartime veterans receiving pensions or disability pay remained eligible for this benefit. That left the families of other veterans with a $150 allowance that must be applied to the cost of a burial plot, not to such other items as mortuary costs or transportation to a gravesite.

VA officials said other cities soon may find themselves with the same problem as Los Angeles: a collection of bodies and a shortage of funds to pay for their burial. Since October, the VA has continued to pay for the burials for those who died before the cutoff date, but by next month, it expects to have verified most of those who died before Oct. 1. Then, county and city governments will be forced to come up with the funds.

Marvin Page, a spokesman for the VA's Veterans Fund Service in California, said the VA has begun assessing the problem of "body held cases" throughout the western United States. But no figures are available from other cities because a report is not expected until early February, Page said.

Robert Kingsbury, director of the Los Angeles County veterans affairs office, said of the situation: "I'm appalled. I think somebody in Washington forgot to do their homework." He predicted the number of indigent vets in the county who will die without receiving the larger burial allowance will reach 1,000 by the end of the year. Los Angeles County has more veterans than any other county in the nation.

The VA was unable to provide statistics on the number of veterans in the country who received burial allowances last year. Nationwide, there are 3.5 million veterans over 65.

Kingsbury said the remains of the indigent wartime veterans would have to be cremated or buried in paupers' graves if the funds were not restored. The latter action, though, would require a change in California law. Dan Wolf, press secretary for Los Angeles County Supervisor Kenneth Hahn, said today that at least one veteran's body was being transported to a local crematorium. But a temporary solution was found when Henry Rushing, a 38-year-old Santa Rosa, Calif., businessman and an Army veteran, provided the $9,000 to bury the man.

"At first I thought it was just the Veterans Administration and the county trying to get more money," he said. "Then I checked it out. I was just brought up to be very patriotic. I grew up in the oil fields. I feel we owe a great deal to the people who fight our wars for us."

Rushing, a self-described Reagan supporter, says he understands the cutback. "I'm not so sure it's a federal responsibility. I don't think it's the government's place anymore to be burying individuals . I agree wholeheartedly with Reaganomics." He said veterans groups, churches and private industry would be better suited to provide for the indigent veterans.

But William Bridges, who has made funeral arrangements in Los Angeles County for thousands of veterans in the past, disagrees. A division chief of the county's office of public administration, Bridges said, "The pattern being set down is that the federal government has pushed the problem down to the local governments. You can ask philosophically if it's the federal government's responsibility to pay the funeral benefits for private individuals. But to pass down the responsibility after carrying it for so many years . . . you can see the problem. We're talking about a significant financial impact."