Secret room in the red wall on Doom 64 level 24, No Escape:

(For video showing how to reach all Secret areas on the map
including the area described below, click here.)

Along one edge of the courtyard is a long red wall, which
runs along the far side of a stream of corrosive slime.
There are two visible doorways along the bottom edge of
this red wall.
A hidden third door exists a few paces to the left of the
other two doors, and can be opened temporarily to reveal a
secret chamber where the BFG 9000 is located.  Here's how
to open that door panel:

Just to the left of the long red wall is a barred doorway.
Behind these bars is the cave where one or more Cyberdemons
are located at the start of the game.
To either side of this doorway are tall columns or pillars.
Step up to the left-hand column and look at it.  Switch to
your wire-frame Automap, and notice that there is a small
yellow line that appears near the surface of that left-hand
column.  This yellow line represents a special switch.  The
switch is located high up on the column, and controls the
secret panel that blocks access to the BFG 9000.
To activate the switch, turn around to face the small
platform on the ground nearby, and walk over to it.
From the platform, walk straight to the large structure
that dominates the center of the courtyard.
Walk up to the raised pillar/lift that's against this side
of the central structure, and Activate it.  When the lift
drops, ride it up and step into the tiny recessed room at
the top.
Turn around to face that column located just to the left of
the doorway to the Cyberdemon cave.
Select your pistol, shotgun or chaingun, and shoot at the
switch buried inside that column.
When you hit the switch, the third doorway opens along the
bottom of the long red wall, allowing temporary access to
the BFG 9000.  Jump down and Run (remember your Run button)
to grab the weapon before the panel closes again.
If you don't get there in time, simply go back to the lift
on the side of the central structure, and shoot the switch
again.
Once you have the BFG, step out of the chamber, turn left,
and Run to the next doorway.  Inside are a few Imps, and a
teleporter...