The German dub of Neighbours (Nachbarn) suffered this.
The French dub of Dallas used this to replace the iconic theme.
The French dub of The A-Team replaced the iconic opening theme tune with a tinny sound-alike featuring French lyrics. The most immediate problem being, the original theme tune doesn't have any lyrics, so there was no need to dub it in the first place. They could have simply recorded the lyrics directly over the original music. To make matters worse, the lyrics are completely redundant, since they simply rehash the premise of the show outlined in the opening voiceover and they refer to the team by the rather less dynamic sounding name of the dub itself, "L'agence tous risques" ("The All-Risks Agency") and also as "The Kings of Cool".
The various Law & Order series had their theme tunes changed in the UK for their Channel Five airings; all the other channels to show them in the UK (BBC1, Sky1, Hallmark/Universal, ITV4) use the Mike Post ones.
When Choudenshi Bioman was licensed in France, it first used a dubbed opening, but did a Channel Hop after five or so episodes and got a new opening by Bernard Minet.
"GO-ON" by Uverworld is the second ending theme for the Japanese dub of the first season and "New World" by w-inds is the ending theme of the second season of House.
The Japanese dub of Prison Break uses "Top Secret" by Namie Amuro for the theme song of the second season. "Evolution" by Exile was the first theme song for the third season and "Be On Top" is the second theme song. "Change" by YU-A is the ending theme song of the final season.
The Japanese broadcasts of Thunderbirds opened with the English-language "5-4-3-2-1" countdown from the original "Thunderbirds March" theme, but then segues into another Barry Gray composition, the Century 21 March, accompanied by Japanese-language lyrics sung by a chorus.
The ending theme for the Japanese release of Damages is "Perfect Me". The ending theme for the second season is "Rain" by Michi.
When RCTV aired Baywatch in Venezuela, instead of using the original opening song, it substituted it with several reggae songs, changing them every season or so. The first and the one used for the longest time was "Bad Boys", the one Americans identify as the theme of Cops.