The Hammers fell to a third consecutive defeat as Sean Dyche's charges scored three unanswered second-half goals at the London Stadium.
Ashley Barnes's opener in the 66th minute prompted a number of supporters to invade the pitch and approach the West Ham players.
One supporter was thrown to the ground by Noble, who was then restrained by goalkeeper Joe Hart, and more fans entered the field of play following each of Burnley's subsequent goals.
"It's well known fans haven't been happy for a long time," the West Ham club captain told BBC Sport. "The only way to squash that is to win games.
"We should have been two up in the first half and as soon as we conceded, the atmosphere changed. The fans' feelings took over.
"I'm a human being. I know I'm out on the pitch, under scrutiny, but if someone approaches me I will protect myself. So will a lot of other players.
"The crowd weren't happy. No matter what happened today, it seemed they were planning something anyway."