Singer Bono said: "We've lost a family member," adding, "He wasn't just a legend in the music business, he was a legend in our band."
Sheehan had been on the road with U2 for more than 30 years.
The band are in the middle of their latest tour and are currently playing a string of concerts in Los Angeles.
They paid tribute to Sheehan on the video screens at Wednesday night's show, said Billboard reporter Shirley Halperin.
As he introduced the song Iris, from their latest album Songs of Innocence, Bono said: "U2 is kind of family. We look after each other and it takes a lot to put on a show like tonight.
"Last night we lost a member of our family. Dennis Sheehan was his name. He was U2's tour manager for 33 years. He loved, as we all do, the city of Los Angeles and he called the Sunset Marquis his 'home away from home'.
"He came to this city as a young man in the 70s, working for Led Zeppelin. He always thought maybe U2 could be the next Led Zeppelin, which of course is impossible."
Sheehan, who was in his late 60s, had worked with other rock legends including Iggy Pop, Patti Smith and Lou Reed.
Los Angeles County Fire Inspector Chris Reade said a medical response team was sent to the Sunset Marquis hotel at around 0530 (1330 GMT) on Wednesday to reports of a man in cardiac arrest.
Sheehan was pronounced dead at the scene.
Global Music Live Nation chairman Arthur Fogel confirmed the death calling Sheehan "a dear friend to us all".
"Our heartfelt sympathy is with his wonderful family," he added.
Sheehan's death is the latest in a series of recent woes for U2.
Last year, Bono underwent extensive surgery following a high-speed bicycle crash, leaving him unable to play the guitar.
Guitarist The Edge then fell off stage on the opening night of the band's tour in Vancouver, gashing his arm.
The band were also derided when Songs of Innocence was automatically uploaded into customers' iTunes libraries.
Bono later apologised, saying the stunt was inspired by "a drop of megalomania".
Irish band U2 have mourned the death of their "irreplaceable" tour manager Dennis Sheehan, who has died of a suspected heart attack.