ELP included the entire Brain Salad Surgery album live on their 1974 Tour.
I was fortunate enough to see Pixies play their album Doolittle, celebrating its 20th anniversary in 2009. It was truly an outstanding show, and they were at the very peak of their reunion form. And while they played the whole album in order, they both preceded that set and then closed out the show with at least an equal number of other songs.
Best part - the album opens with a track called Debaser, which makes reference to the 1929 Buñuel film Un Chien Andalou, a film, which famously has a horrific (special effects) scene of a man calmly cutting an woman’s eyeball open with a razor. They opened the show by playing that scene on the screen behind the stage. Us old folks all knew to look away or turn around. The young kids in the crowd were caught completely by surprise for the most part. Their reaction was amazing.
About a year ago I saw Styx play their entire album “Grand Illusion” in concert. Then after a break they played the hits from their other albums. A very satisfying show.
I’ve seen several play throughs. The best part about them is when they include tracks you love but seldom hear. Even Coheed and Cambria playing In Keeping Secrets of Silent Earth 3 wasn’t an experience of the album as a whole even though they are the proggiest of the playthroughs I’ve seen. The best part was definitely hearing The Light and the Glass for only the second time in more than 10 Coheed concerts I’ve been to.
Ditto for Get Up Kids playing Something to Write Home About, although I’ve only seen them twice before so I don’t know if they rarely play tracks other than 10 Minutes to Downtown and I’m a Loner Dottie, a Rebel.
Actually come to think of it, the best “cohesive album” playthrough moment for me was when I saw Say Anything play … Is a Real Boy. The first 4 tracks really work well together even though the band isn’t prog as a whole.
And then there’s the negative experience of when I saw Saves the Day play Stay What You Are. I wasn’t expecting it and it wasn’t that great since I don’t really like the deeper cuts (or even the concert staple of Cars and Calories, although I do really like the concert staple of At Your Funeral.) I would have preferred them play their other songs instead. Although it was satisfying that I could tell they were playing the album even though I didn’t know that they were going to.
A few times. But a couple of them are probably cheating.
I was there for the live album that Alison Krauss and Union Station recorded in 2002. Since it’s live, it was a combination of different albums and projects (like some of the songs from O Brother Where Art Thou), but it’s a complete album that was released, so I technically saw them perform that full album that was later released.
I saw Queensryche several times when they were performing Operation: Mindcrime. But that’s a concept album, so they kinda had to perform the whole thing.
Those are the two I consider cheats.
The time I saw Ben Folds, it was after he and the band dissolved. He’d just come out with his first solo record, Rockin’ the Suburbs, and performed the entire album.
And two of the times I’ve seen Counting Crows, they had just released new albums, and performed them in their entirety. The first time was Recovering the Satellites, and I saw them later on right after This Desert Life was released.
Never got to see the Who perform Tommy, but Roger Daltrey and his solo band (which is basically the rest of the Who’s current backing band aside from Pete Townshend) toured with Tommy in 2011 and I got to see that. I also got to see Rush do Moving Pictures on the above-mentioned tour, and Roger Waters doing The Wall in either 2009 or 2010.
Green Day did a tour two years ago where they played two Albums back to back - starting with Dookie for its 30th anniversary, then 4 individual songs, then American Idiot for its 20th anniversary.
I saw Boston in 1987, when they were touring behind their long-delayed third album, Third Stage. They opened with a few of the best-known songs from their first two albums, and then played every song from Third Stage, in the order in which the songs appeared on the album.
I saw Peter Frampton on his “Frampton Comes Alive! 35” tour in 2012, which was for the 35th anniversary of his famous live album. He played every song from “FCA!”, though not in the album order, as well as a lot of additional songs.
On a tangential note, I saw the Reduced Shakespeare Company do a show at a historic theater in Tacoma’s Theater District in 2012, on the same night that Frampton was playing in another historic theater directly across the street. Before the show started, the emcee made a point of announcing “Just so we’re clear, if you were expecting anything to Come Alive tonight, you’re in the wrong building,” and the RSC then made several jokes at the expense of audience members who weren’t laughing that they must still be waiting for Frampton to come on.
Sparks went further with this than anyone ever:
In 2008, they performed their (then) 21 albums live in their entirety, from first to last, in 21 consecutive nights.
I don’t remember having attended a show where a whole album was played, but I got the live albums from the tours when Van Morrison played “Astral Weeks” in its entirety and Lou Reed “Berlin”, both about 40 years after the original albums had been released.
ETA: Belle And Sebastian touring with “Tigermilk” and “If You’re Feeling Sinister” came close to me (to Cologne on two consecutive days), and since they’re also one of my favorite bands (and those two albums are among my favorites in their catalog), I considered going, but in the end it wasn’t in my budget. But I believe that it must’ve been great fun, I’ve seen them around five times, and Stuart Murdoch always likes to comment on or joke about songs at stage. He’s a very witty guy.
I saw Social Distortion play their whole eponymous album for its 30th anniversary. A great show.
Playing an entire album live is common in prog rock, especially when a band is promoting a concept album. (A real concept album, that should be taken as a single piece of music; not a collection of songs that are loosely connected.) I have tons of live albums with entire albums being played, but sadly I can’t think of any I’ve personally seen. The only entire album I can think of that I’ve seen performed wasn’t even prog: Quadrophenia in '96 or '97.
How about a band playing someone else’s album in its entirety? 20 years ago and more, Dream Theater would sometimes come out for their encore and cover an entire album. Several of these have been released as “unofficial bootlegs:”
Master of Puppets
The Number of the Beast
Dark Side of the Moon
Made in Japan
I saw The Who on the original Tommy tour. First they played a 45 minute set of their other material. That was a bit longer than headliners usually played. Roger said something to the effect, “We were supposed to take a break than come back for Tommy, but we’re too pumped.” Then they played Tommy in its entirety.
(to be continued)
Dammit! I knew they had misspelt the title but still put the accepted spelling! 
I think you chose the wiser path, imagine how many response correction posts you might’ve gotten if you had spelled it the way they did.
Though they are older albums, the recent lineups of the prog band Yes have done a number of tours over the last decade or so in which they played one or more of their classic albums in full.
Steve Hackett of Genesis has done the same thing with his “Genesis Revisited” shows. I saw him on his “Foxtrot at 50” tour a few years ago where they played the entire Foxtrot album.
I saw Cheap Trick perform Sgt. Pepper’s Lonely Hearts Club Band at the Hollywood Bowl about 20 years ago.
Whew. My brother is a prog fan and I find prog to be exhausting. Seeing a whole show and then the second half of the show is someone else’s entire album? Good grief! I have no patience for these people 
(seriously tho, I’m sure it’s cool to fans)