Stewards raced over to an area of the East Stand at Brisbane Road, with supporters desperately trying to get the match stopped.
The match went on for a few minutes before one fan ran onto the pitch to try and get the attention of the referee.
Around 20 then entered the pitch and sat by the touchline in a bid to get the match paused.
Orient boss Richie Wellens was spotted walking over to the supporters demanding they get off the pitch, seemingly unaware of what was going on.
The match was then paused, with the home side leading 1-0 at the time.
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Lincoln City goalkeeper Lukas Jansen reportedly refused to restart the match while medical staff dealt with the emergency.
An announcement was made at 10:52pm that the game had been abandoned, nearly an hour after the match had been stopped.
"Tuesday's fixture against Lincoln City was abandoned due to a medical emergency in the East Stand of Brisbane Road," an Orient statement said.
"The thoughts of everyone at Leyton Orient Football Club are with the individual involved and their loved ones at this distressing time.
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"The club would like to thank all supporters in attendance at Brisbane Road this evening for the understanding and compassion shown, during a very difficult situation.
"The club will provide a further update in due course."