This could be a bug in Perl or your not using the correct syntax to handle the variable.
Here is my fix to your code.

#!/usr/bin/perl

my $code1='my $var_with_é_accent;';

print "Eval buggy code\n";
eval {$code1;}; # Should see "Useless use of private variable in void 
+context at Untitled line 6."
print "ERR = $@" if ($@);; # Error doesnt show here
print "DONE\n";

my $string;
my $code2='$string="new eval done";';

print "Eval good code\n";
eval $code2;

print "ERR = $@" if ($@);
print "string is : $string\n"; # This Works
print "DONE\n";
[download]

Good Luck ^^