# THIS WORKS
print (OUT $line) || die "writing 'splat' $!";

# But putting || die on another line does not. i.e:
# print (OUT $line) 
#       || die "writing 'splat' $!";
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That's strange. They both work for me.

Moreover, I see no reason why they would be different, in fact Deparse tells me they produce identical code:

% cat id1.pl
 print (OUT $line) || die "writing 'splat' $!";

% perl -MO=Deparse id1.pl
die "writing 'splat' $!" unless print OUT $line;
id1.pl syntax OK

% cat id2.pl
 print (OUT $line)
    || die "writing 'splat' $!";

% perl -MO=Deparse id2.pl
die "writing 'splat' $!" unless print OUT $line;
id2.pl syntax OK
[download]

Though you could dig further into the root cause of the difference you experienced, I suggest you just stick to the low precedence or and avoid parens with print as described earlier.