IO#dup will copy a file path.

Test Script:
{{{
require 'test/unit/assertions.rb'
include Test::Unit::Assertions

io = STDOUT.dup
assert_equal("#<IO:<STDOUT>>", io.inspect)

io = File.open("/tmp/foo", "w")
io2 = io.dup
assert_equal("#<File:/tmp/foo>", io2.inspect)
io.close

r, w = IO.pipe
io = r.dup
assert_match(/#<IO:fd \d+>/, io.inspect)

f = File.open("/tmp/foo", "w")
w.reopen(f)
assert_equal("/tmp/foo", w.path)

puts :ok
}}}

io.c

@@ -2735,6 +2735,9 @@ io_replace_streams(int fd, rb_io_t *dest, rb_io_t *origin)
 	dest->buf = NULL;
     }
     dest->buf_offset = origin->buf_offset;
+    if (origin->path) {
+	GC_WB(&dest->path, rb_str_dup(origin->path));
+    }
 }
 
 static VALUE
@@ -2770,10 +2773,6 @@ io_reopen(VALUE io, VALUE nfile)
     }
     io_replace_streams(fd, io_s, other);
 
-    if (other->path) {
-	GC_WB(&io_s->path, rb_str_dup(other->path));
-    }
-
     *(VALUE *)io = *(VALUE *)nfile;
 
     return io;