I tried burning CD-Rs with a SONY CRX145E CD Burner and other CDR-s with a
SONY DRU510A burner, but CDR disks overheat when played back on my Panasonic
CQ-DP383U car CD player.

This morning I played a pressed CD, it didn't overheat. The problem seems
only to occur when I playback CDRs.

It shouldn't be a problem with a CD-R. CD-RWs maybe, but not CD-Rs. What
burner are you using?