A federal District Court judge rejected pleas today that would have kept Henry Grady Gore Jr., a member of a politically prominent Washington family, from serving a one-month jail term for non-payment of income taxes.

Judge Alexander Harvey ordered the 61-year-old Gore to begin serving his sentence at the Montgomery County Detention Center immediately.

Gore managed the family-owned Fairfax Hotel at 21st Street and Massachusetts Avenue NW in Washington for more than 20 years until he was indicted on the tax charges last April. His sister, Louise P. Gore, ran unsuccessfully against Marvin Mandel as the Republican candidate for governor in 1974.

As part of a plea bargaining agreement with the government, Gore pleaded guilty last April to non-payment of taxes on an income of $61,339 in 1970. Government prosecutors dropped charges of non-payment of taxes for 1971 and 1972.

Last June Judge Harvey sentenced Gore to one month in jail, 23 months probation and a $55,000 fine.

Gore's attorneys, James P. Parker and Lee Monroe, said his tax problems resulted from an unsuccessful battle with acute alcoholism and argued that a recent voluntary hospital confinement should substitute for the jail term.

The judge said there is no legal precedent for substituting hospital confinement for a jail term, and that "His alcoholism was considered in imposing the sentence."