As far as I understand what is happening in that scene, the red pill fakes a death: From the script:
The mirror creeps up his neck as Neo begins to panic, tipping his
head as though he were sinking into the mirror, trying to keep his
NEO It's all over me --
Morpheus is right next to him, with the phone.
TRINITY I got a fibrillation!
CYPHER I knew it, I knew it...
MORPHEUS Shit! Apoc?
Streams of mercury run from Neo's nose.
APOC Targeting... almost there.
An ALARM, on Trinity's monitor ERUPTS.
TRINITY He's going into arrest!
CYPHER He's gonna pop!
APOC Lock! I got him!
MORPHEUS Now, Tank now!
His eyes tear with mirror, rolling up and closing as a high-pitched
ELECTRIC SCREAM ERUPTS in the headphones --
It is a piercing SHRIEK like a computer calling to another computer
Neo's body arches in agony and we are pulled like we were pulled
into the holes of the phone
Sucked into his SCREAM and swallowed by darkness.
I always thought that the pill simulated a heart attack (the fibrillation signal may be helping them to find the body), so the system just disposes him in order to be recycled. There is no need to kill a dead body.