BORROWING A PAGE from the book of Marion Barry, Chicago Mayor Jane Byrne decided last weekend that The Chicago Tribune could no longer use its press facilities in city hall. The mayor announced in a press statement that "The Chicago Tribune has engaged in innuendos, lies, smears, character assassinations and male chauvinist tactics since Jane Byrne became mayor," and that therefore she was tossing the paper out.So there.

Our very own Mayor Barry, perhaps recognizing that anyone assigned to earn his living by getting information out of the District Building already has been punished enough, has reacted to what he calls "negative" stores in the press by keeping the press room open. Instead, he has closed down the press office in his mayoral suite and dispatched his erstwhile press secretary elsewhere. Mayor Byrne could not very well do that; not even Mayor Byrne. Her press secretary, it turns out, is none other than her husband, Jay McMullin, a former city hall and real estate reporter on leave from The Chicago Sun-Times to bail out Her Honor.

Now, Chicago politics, we are told, are unlike those in any other city, which is somehow supposed to serve as a justification for the way the City that Works actually works. But even in a city accustomed to outrageous politics, Mayor Byrne is making quite a name for herself.

She seems, for example, to be forever changing her mind. At first she's going to back President Carter. Then, at the last moment, she throws in her lot with Senator Kennedy. Then she can't deliver. She changes her mind about her aides and her policies, and has gone through so many high-level assistants that the papers have taken to calling her administration "government by revolving door." She had four police commissioners in her first 10 months. Her husband is her third press secretary.

What set her off last weekend was a story in The Tribune detailing waste and inefficiency in the Chicago city government before she took office. The story was based on a report by her transition team and noted that she had not followed many of the team's recommendations. Why the mayor got so steamed about a report detailing conditions that weren't her fault is a mystery, but the upshot of the story was that the mayor ordered the Tribune reporter out. At least that's what seemed to be going on. By Monday, the Tribune reporter was still in city hall, the mayor had not evicted him, her husband was calling him a "squatter," and the Tribune had another banner headline: "Byrne Backs Down, Tribune Stays."

Mayor Berne, again like our very own Mayor Barry, has not had an easy go of it. She's had to tackle allegations of mob influence in the police department and city hall. Barry had to face a budgetary nightmare. If she has trouble keeping track of her appointees, the Barry administration has had trouble keeping track of the city's foster children. While both mayors promised an open administration, their reactions after more than a year in office have been to open fire on the press.

It seems that Mayor Byrne and Mayor Barry have decided that the press is the root of municipal evil. If the press weren't guilty of all its lies and distortions, of its "negative" reporting, their cities -- not to mention their Honors -- wouldn't have nearly as many problems as they do.

They aren't the first mayors to discover that a certain institutional tension arises when their administrations aren't working very well and the press is. But Mayor Byrne and Mayor Barry have added a new dimension to that tension. She comes right out and says it. Her critics in the press are guilty of "male chauvinist antics." Mayor Barry is a little more subtle.He takes black reporters aside and warns them that they should not let themselves be manipulated by their white editors. Some reporters who have heard it call it The Speech.

The idea, of course, is to obscure and intimidate. You, the mayor, are getting picked upon by the press, not because you are doing a bad job, but because you are a woman or a black or whatever ethnic or religious minority you can fall back on. Mayor Byrne has been going on about her Male Chauvinist Pig critics for some time now, all the while behaving in a way that should be the absolute delight of every eligible MCP voter in sight. Who else has done so much to resurrect the stereotype of an erratic menopausal female office holder who -- just like a woman -- can't make up her mind?

It is all too easy for thin-skinned politicians like Mayor Byrne to answer critics by labelling them male chauvinists or whatever prejudice is appropriate to the occasion. Of course it doesn't solve the city's problems and it is demeaning to the voters to think they are satisfied with that kind of answer for poor performance, but no matter. If it works, it gets you off the hook. Or at least it gets you sympathy.

But it doesn't work with Byrne. It doesn't take a male chauvinist press to figure out that the lady simply is not acting like a mayor.