A CD player that otherwise works perfectly won't play a certain CD burned
from an iTunes playlist. But this particular CD works perfectly in every other
player I've tried. What could be the hangup?

Is it a CD-R or some other format like a -RW?  Have you tried
re-burning it?  Do subsequently burned CDs work, as well as previously
burned CDs?

Try making a copy of this CD and see if that will play.
Perhaps there's unnoticeable scratches on this CD?

I am going to assume it is a CDR or CDRW since you burned it so
therefor it must be.  In this case, surprisingly, not all CD players
will play it.  This is because different companies structure the bumps
in grooves in different fashions.  The particular CD player will only
be able to read certain brands of CDRs and the higher quality it is
(or newer) will determine how many types it can play.  For Example My
92' Pontiac Grand Prix wasn't designed to play CDRs yet it could play
about half that I tried but my computer could obviously play them all.
    CDRWs will have a harder time working in any cd player.  If that's
not the case then it may be the files were not converted before they
were burned which would be strange since Itunes does this
automatically.  In this case you would need to change your preferences
to convert them.  Go to Edit::Preferences and choose the 'burning'
tab.  Now for disc format click audio CD.  That's really all that
could go wrong since it worked on other machines and possibly they
were MP3 CD players.

A problem that sometimes occurs with old CD players and cheap
CD-R/CD-RWs is that they can actually slip, try lining the inner hole
on the cd with sticky tape (1 layer is all that would be needed). It
doesn't happen as much as it used to when blank CD's first became
available, but might be worth a try as a quick, cheap fix.

Thanks,
Gareth