The differences you see come about because or takes precedence over if. Because of this, or doesn't act like "else". Instead it turns its neighbors into a single condition used by if. If you put in parenthesis to explicitly show precendence, the two statements you wrote above look like this:

print "TRUE" if (0 or die "FALSE");
print "TRUE" if (0 or print "FALSE");

#or alternatively

if (0 or die "FALSE") {
   print "TRUE";
}
if (0 or print "FALSE") {
   print "TRUE";
}
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Once you see the precedence, the difference in behavior is obvious: when 0 or die "FALSE" gets evaluated it dies before ever getting to what is inside of if {...}. On the other hand, 0 or print "FALSE" evaluates to true and causes Perl to enter inside of the if {...} statement. Hence, "FALSETRUE" gets printed.

Best, beth