The MAP is used during warm-up, as well as post warm-up. I think it's used at all times; but not sure if it's used during wide open throttle--the ECM usually has a special "program" for wide open throttle conditions. Generally, the ECM has an internal timer that holds off for ~1 minutte before allowing closed loop operation. So on a warm engine restart, with the coolant already above 150*f, two things must occur before going closed loop: The timer must elapse, and the O2 sensor must start sending a varying signal to the ECM. If the ECM doesn't see the O2 signal rapidly varying between approximately .2 and .7volts, it won't go into closed loop, but will continue to adjust fuel based on engine temp and MAP input. For the O2 sensor to operate, it must reach its operating temp of arround 600*f. I beleive all XJ's have O2's with a heater circuit to insure operating temp is maintained, but it works in conjunction with exhaust gas heating (some systems don't use an O2 heater circuit--my engine Chevy doesn't).
Why don't you share with us what your experiencing?
Basically, a cold start will have a richer fuel mixture than a warm, restart; this is a direct result of the engine and MAT sensors. Don't no about non-Renix ECMs, but early XJ's always had an idle "flare-up", even on a warm restart--the ECM always opens the AIC at engine shut down, in anticipation of the restart. This increased idle should be very short lived after a warm start, as the engine temp interacts with the ECM to control idle speed; warm engine=low idle--just like in the olden days of carbs with automatic chokes.