Sarri’s long-term future at Chelsea is in doubt despite only being appointed last July.
The Blues have slipped to sixth in the Premier League table and club chiefs discussed potential successors to the Italian last week.
Matters were not helped by losing the Carabao Cup final to Manchester City on Sunday.
The penalty shootout defeat will be overshadowed by goalkeeper Kepa Arrizabalaga refusing to be substituted in extra time.
Burley says the incident shows there is no leadership on the pitch, as well as off it.
And ahead of the Premier League visit of Tottenham on Wednesday, the ESPN FC pundit has has painted a bleak picture for the Chelsea boss.
“There’s no leadership at this club,” Burley said.
“There’s no leadership in the structure, there’s no leadership on the pitch, and we talked about that because there’s no captain.
“Sometimes there shouldn’t have to be if you’ve got strong enough characters elsewhere, like going and dragging the keeper off.
“But, you know, [Cesar] Azpilicueta, who was captain on the day, says: ‘Well I was over there on the other side, it’s not my responsibility’.
“So there’s just no leadership and when there’s no leaders as players, that has to come then from the coaching staff and the manager.
“He’s not leading at the moment, he’s not, he’s caught in the headlights.
“I don’t know if it’s because he’s never faced these scenarios before – he’s managed for quite a while.
“But he’s struggling with the concept of managing this club at the moment and these players, no doubt about it.
“He’s a dead man walking, no matter what happens, he’s a dead man walking.”
Marcos Alonso could return to the Chelsea squad against Tottenham after being left out against City on Sunday.