Sorry for the delayed reply.

In the examples below, I used the following test harness:

my $module = do {
   open(my $fh, '<', 'Module.pm')
      or die("Unable to read Module.pm: $!\n");

   local $/;
   <$fh>
};

eval $module or warn $@;
$module =~ s/^for;$/#for;/m;
eval $module or warn $@;
[download]

• The most common problem would be (safely ignored) warnings.

  use strict;
  use warnings;
  
  package Module;
  
  sub one { }
  for;
  sub two { }
  
  1;
  [download]

  First require/eval output:

  [None]

  Second require/eval output:

  Subroutine one redefined at (eval 2) line 6.
  [download]

• Another problem is code running twice.

  use strict;
  use warnings;
  
  package Module;
  
  END { print("END1\n"); }
  for;
  END { print("END2\n"); }
  
  1;
  [download]

  First require/eval output:

  [None]

  Second require/eval output:

  END2
  END1
  END1
  [download]

• Same, but different.

  use strict;
  use warnings;
  
  package Module;
  
  BEGIN { print("BEGIN1\n"); }
  for;
  BEGIN { print("BEGIN2\n"); }
  
  1;
  [download]

  First require/eval output:

  BEGIN1
  [download]

  Second require/eval output:

  BEGIN1
  BEGIN2
  [download]

• Objects can be created and destroyed twice.

  use strict;
  use warnings;
  
  package Module;
  
  {
     package Obj;
     sub new     { print "Creating\n"; bless({}, shift) }
     sub DESTROY { print "Destroying\n"; }
  }
  
  our $o;
  
  BEGIN { $o = Obj->new(); }
  for;
  
  1;
  [download]

  First require/eval output:

  Creating
  [download]

  Second require/eval output:

  Subroutine new redefined at (eval 2) line 8.
  Subroutine DESTROY redefined at (eval 2) line 9.
  Creating
  Destroying
  Destroying
  [download]

  Hopefully, nothing took the address of $o in between the two requires!

That's all I can think of. It's not quite as bad as I thought, since I thought a function could capture a variable from the first attempt, and another function could capture the same variable from the second attempt. Still, the above can still create some weird problems.