Why did Catherine Petkoff lie to Major Petkoff about his old coat?

There are two major reasons. The first, and most pressing, is that Catherine doesn't want Sergius, who is present, to hear the truth about Bluntschli.

Telling the truth about the coat would mean revealing that Bluntschli had visited Raina in her bedroom, and this would likely end the engagement between Sergius and Raina. Catherine notes this explicitly in an aside to her daughter:

If Sergius finds out, it will be all over between you.

And we see later in the play, Catherine's fears were justified. When Sergius discovers that Bluntschli had visited that night, he is quick to interpret this as evidence that Raina has been "trifling with another man behind [his] back" and he is ready to denounce her.

Second, telling the truth would mean confessing that she and Raina were the anonymous women depicted in the story that Sergius had recounted to them. Even if there had been no engagement between Sergius and Raina, Catherine wouldn't have liked this revelation to come out. She is very concerned with her image as a dignified, cultured, high-status woman. In Sergius's gossipy account, the women who sheltered Bluntschli had shown a lack of aristocratic judgment. The women are depicted in a way that makes them seem amusing or ridiculous. They were credulous, easily flattered and taken in by a lower class man unworthy of their esteem:

SERGIUS [with bitter irony] Oh yes: quite a romance! He was serving in the very battery I so unprofessionally charged. Being a thorough soldier, he ran away like the rest of them, with our cavalry at his heels. To escape their attentions, he had the good taste to take refuge in the chamber of some patriotic young Bulgarian lady. The young lady was enchanted by his persuasive commercial traveller's manners. She very modestly entertained him for an hour or so, and then called in her mother lest her conduct should appear unmaidenly. The old lady was equally fascinated; and the fugitive was sent on his way in the morning, disguised in an old coat belonging to the master of the house, who was away at the war.

Note the derisive tone, and the digs at everyone concerned. The soldier "ran away," which is inconsistent with the idealized image of a gentleman that Sergius and his cultural circle favor. The young woman entertained the soldier for an hour before deciding -- too late, by this same cultural standard -- that she needed a chaperone. And the mother, described as an "old lady" (which can't have pleased Catherine), is equally credulous.

The stage directions suggest that every compliment paid to the characters is ironic, indicating that Sergius really means to say the affair wasn't a heroic romance, but something ridiculous and low. The women were silly to have treated the soldier as a gentleman worthy of such hospitality, and their actions suggest they are not true elites. True elites would have known better.

So Catherine doesn't want to be identified in this way. It would open her to derision on a point that is dear to her self-image.