What year did they begin announcing that TV shows were "Filmed in front of a live studio audience"?

One of the earliest, if not the first such announcements was "All in the Family was recorded on tape before a live audience" from 1971.

According to Michael Tueth's Laughter In The Living Room (2005), one of the earliest American sitcoms to broadcast live was I Love Lucy (1951 – 1957). For personal reasons, Lucy Ball wanted to produce the programme from Hollywood instead of New York City, but New York was the only place where live TV was produced and broadcast live. Ball suggested they film in Hollywood, and broadcast from New York. This started a trend.

Tueth says they also insisted on filming with a live audience, but this wasn't repeated until a new wave of comedies in the 1970s, such as All in the Family (1971 – 1979) started announcing it "was filmed in front of a live studio audience".

Wikipedia says:

All in the Family was the first major American series to be videotaped in front of a live studio audience. In the 1960s, most sitcoms had been filmed in the single-camera format without audiences, with a laugh track simulating audience response. Lear employed the Multi-camera format of shooting in front of an audience, but used tape, whereas previous multi-camera shows like Mary Tyler Moore had used film. Thanks to the success of All in the Family, videotaping sitcoms in front of an audience became common format for the genre during the 1970s. The use of videotape also gave All in the Family the look and feel of early live television, including the original live broadcasts of The Honeymooners, to which All in the Family is sometimes compared.

For the show's final season, the practice of being taped before a live audience changed to playing the already taped and edited show to an audience and recording their laughter to add to the original sound track. Thus, the voice-over during the end credits was changed from Rob Reiner's "All in the Family was recorded on tape before a live audience" to Carroll O'Connor's "All in the Family was played to a studio audience for live responses". (Typically, the audience would be gathered for a taping of One Day at a Time, and get to see All In the Family as a bonus.) Throughout its run, Norman Lear took pride in the fact that canned laughter was never used (mentioning this on many occasions); the laughter heard in the episodes was genuine.

Also from Wikipedia:

Creator Norman Lear's All in the Family (CBS, 1971–1979) followed suit in 1971. Videotaped live, Lear wanted the studio audience to actually like the performer, with hopes of the two developing a rapport with each other. Lear was not a fan of pretaped audiences, resulting in no laugh track being employed, not even during post-production when Lear could have had the luxury of sweetening any failed jokes (Lear relented somewhat in later seasons, and allowed [sound engineer and laugh track inventer Charles] Douglass to insert an occasional laugh). Lear's decision resulted in the show being a huge success, and officially ushered in the return of live audiences to the U.S. sitcom mainstream. To make his point clear, an announcement proclaimed over the closing credits each week that "All in the Family was recorded on tape before a live audience" (or during the show's final seasons where live audiences no longer attended tapings of the show) "All in the Family was played to a studio audience for live responses."

The announcement of "filmed before a live audience" most likely started with the The Odd Couple, in 1971. The reason why is that The Odd Couple started out being filmed on a closed set in the first season. Then in the second season, it completely changed format and was shot in front of a live studio audience. Because it was such a radical change in format, it was probably felt that it was necessary to alert fans of the show that it was no longer being shot on a closed set, like when back in the 1960s, shows that started out in black and white and transitioned to color also had an announcement before they started that they were "now in color."