I have a 1988 DL 5-speed with rebuilt engine and new clutch.  Recently when I shift into 5th gear a high-pitched metallic screech occurs.  It goes into gear but if I just ever so slightly push on the shifter the noise happens.  What is it?

I would venture to say you have a synchro problem if it only occurs during the shift and is not constant in 5th gear. Might also be the blocking ring. The only way to know is to tear down and inspect. If it'f that audible, you're going to see damage to some part around the 5th gear set when you look. Depending on what it is, it could go for years or take out bearings or other blocking rings.

Thank you skishop69 for your advice.  I too and others thought that the synchro gears may be worn but am happy to report that the transmission was not the culprit after all.  It turns out that my drive line was worn and tired and rotating ellliptically thus hitting against my shifter linkage - especially in 5th gear.  There are three staked (non-greasable) u-joints with one carrier (hanger) bearing on the drive line.  I took the drive line to a drive line specialist.  The drive line is original to the car with 285,000 miles.  The specialist told me that all three u-joints were too far gone plus the shafts were also too far gone because the yoke ears were badly degraded and worn and would not properly house new replacement u-joints.  So I went to a junkyard and found a replacement drive line for $25 and took it to the specialist.  He pronounced that the new drive line was in rather good condition with only one of the three original staked u-joints needing replacement.  He used an greasable aftermarket u-joint RP 430-10.  The part cost $31 and labor to remove the staking and install was $5.  The car runs smoother and quieter. 

Is it a dual range transmission or a push button? There is another, simpler, possibility (though SLCPUDs is more likely).

I've had many an EA82 that did this. There's a bushing for the shifter linkage that wears out and allows it to sag down onto the driveshaft.

Tell me more about this bushing.  Now that my car is quieter I now hear an annoying faint rattle in 3rd and 4th gear.  Is this the bushing that I can see from inside the car by pulling up the boot? 

You might be able to, but you can definitely see it by getting under the car. Actually, I believe I zip-tied a piece of wood in place of the rubber and that worked just fine.