The Internal Revenue Service, whose tax-processing operations last year were fouled by computer error and huge backlogs of unprocessed returns, said yesterday that it has approved more than 12.5 million refunds and that operations have been trouble-free so far.

The number of refunds approved as of March 7 was 48 percent higher than the 8.5 million refunds approved by the same date last year. Aided by 20 new computers, 4,900 additional workers, improved training and 336 new toll-free telephone circuits nationwide, the IRS has processed 20.97 million returns. That figure is up 40 percent from the 14.9 million returns the IRS had handled by the same date last year.

The Philadelphia Service Center, which handles tax returns from the District and Maryland, had major problems last year. This year, however, processing appears to be going smoothly. Compared with 589,000 returns processed by this time last year, the Philadelphia center has completed work on 1.7 million returns this year.

"There are no significant processing problems in the Philadelphia center and we expect no significant processing problems," said Joseph Cloonan, the center's director.

The Memphis IRS center, where Virginia residents' tax returns are handled, has processed 2.7 million returns so far this year, compared with about 2 million at the same time last year.

"We certainly feel last year's problems are behind us," said IRS spokesman Larry Batdorf. "Everything is going smoothly." He said that the service had paid $49.5 million in interest last year, including interest on tax refunds that were delayed -- considerably more than the $28.9 million in interest the IRS had paid the year before.

"So far, so good," said Robert M. Tobias, president of the National Treasury Employes Union, which represents IRS workers. But he added, "The potential for getting sidetracked, however, is very large. There are lots of things that have to stay in place for the processing to continue as well as it has."

One factor that may have helped this year is that returns have been slower coming in. In January, the number of returns received was running about 19 percent behind the number received at that point in 1985. The service had received 39.6 million returns as of last Friday, down approximately 2 percent from the 40.37 million returns it had received by the same date last year. However, the number of returns received last week was higher than for the comparable week a year before, Batdorf said.

Last year, the IRS was struggling to make a new computer system work and running far behind in processing returns and answering correspondence -- including inquiries from taxpayers about why they had not received their returns.

Tobias said that he has noticed lately that the backlog of unanswered correspondence is increasing, "and whenever that happens, the problem increases geometrically because people start calling on the phones and you have to have more people assigned to problem resolution." Still, he said, "we're certainly in better shape this year than we were at this time last year."

From the perspective of the office of Sen. John Heinz (R-Pa.), things are also going better. "So far, they've been processing more returns and getting out more refunds," said Richard Bryers, Heinz's press secretary. "It's head and shoulders above last year."