I’m not sure if duos are allowed. Obviously, they are much more likely to appear irreplaceable. If you lose one of the members of Sonny & Cher, how could you replace that? Nevertheless here’s a few more recent rock groups, mostly duos, where I believe the band would cease to function if a member was lost. In the case of the duos I tried to pick ones that at least appear a bit more bandlike than say - Rodrigo y Gabriela or Simon and Garfunkel.
The Kills - a sort of slick bluesy sex rock duo, both are completely irreplaceable. U R a Fever
Yeah Yeah Yeahs - art rock trio comprised of vocals, guitar, and drums. Each respected in their own right, each with a unique sound and take on the instrument. Maps
Two Gallants - a sort of americana indie southern rock duo, the vocalist/guitarist has a unique voice and fingerpicking style while the drummer is very start/stop (I’ve heard it described as melodic rather than rhythmic drumming, take that with six grains of salt) - neither could be replaced without creating a totally different sound. You Losin’ Out
The Dodos - Indie rock band where the singer/guitarist fingerpicks over a variety of tunings, complemented by a drummer who studied both metal patterns and african hand drumming. Red and Purple
You can’t replace anyone from The Police. The case has already been made for Andy and Stewart, so I’ll speak for Sting. By playing reggae style bass lines, he allowed sonic room for the other gents to go crazy. They had to fill the space between Sting’s bass notes. I think I’m remembering this line of logic from an interview with one of the guys, but I can’t be sure. I believe it, anyway. Walking on the Moon