A federal judge, seeking to end "bureaucratic disputes" among District of Columbia government agencies, ordered the city public school system yesterday to provide free around-the-clock care for a 16-year-old emotionally disturbed, handicapped youth.

The order, by U.S. District Court Judge Harold H. Greene, brought to light another of an increasing number of troublesome legal and governmental disputes centering on the educational problems of emotionally disturbed, retarded and other handicapped youngsters, according to several lawyers familiar with the issues.

The court ruling also pointed to an apparent rift between two major District agencies, the Board of Education and the Department of Human Resources.Greene criticized the two agencies for "seeking to shift the responsibility to each other" for educating and caring for the youngster, who had been diagnosed as a multiple handicapped epileptic.

And Greene's order also told the wrenching tale of a Northeast Washington couple and their protracted struggle to obtain adequate care for one of their several children.

The Washington Post will not identify either the youngster or his parents at the request of legal and medical authorities involved in the case.

"The saga of the unsuccessful efforts of [the family's] counsel to secure action from the District of Columbia agencies is long and tortured," Greene said in his order. "All during December, counsel contacted various officials of the D.C. Board of Education and the D.C. Department of Human Resources, each of whom disclaimed responsibility and failed to achieve an appropriate placement [for the youngster]."

According to Greene's order, city school officials initially evaluated the boy in November 1977 and recommended placing him in a 24-hour-a-day facility where he could get medical, psychological and educational care. No further steps were taken, however, until an order was issued three months later by a school hearing officer, a judge's ruling noted.

In April 1978, the youth was sent to a Pennsylvania institution, the Elwyn Institute, under a city contract to provide special education for him. By June, however, problems developed and last December the institute notified his parents, a couple of modest means that he would be discharged because the school could no longer cope with his emotional and other problems, Greene's order said.

Then began a new effort by his family to find a suitable way for caring for the youngster. Eventually, he was discharged by the school and assigned to a mental health unit at D.C. General Hospital. Later, he was transferred to the Dominion Psychiatric Treatment Center in Falls Church, an educational and theraputic institution.

Under Greene's order yesterday, the youth must remain, at least temporarily, at the Dominion Center. But in one significant shift, Greene ordered the D.C. school system to pay the costs of the youth's care - about $160 a day.

Until now, the family had paid about $130 of this daily fee through the father's health insurance coverage, according to the family's lawyer, J. Dennis Doyle, a staff member of the Urban Law Institute of Antioch Law School. The remaining $30 a day was paid by DHR, the city's social services agency, Doyle said.

City government lawyers declined to comment on Greene's order yesterday, saying they had not had time to review it. But Doyle praised it as a significant ruling, asserting that increasing numbers of handicapped youths have been "falling through the crack" between DHR and the school system with nowhere to turn but to the courts.