1. view the source. Line 522. Do you see a likely path to fixing it? If so then move on to
2. Check the Repository section of the documentation for a pointer to the best way to download the source.
3. Check the repo to see if there are newer changes that haven't been pushed. (there aren't).
4. clone the sources from github: http://github.com/Corion/mozrepl-remoteobject, patch and push back to author.

git clone http://github.com/Corion/mozrepl-remoteobject
cd mozrepl-remoteobject
make test 
# make changes
# ...
[download]

It looks like the test uses MozRepl::Plugin::JSON2 and sends a unicode string through the pipeline to see if it comes out cleanly. It doesn't. So either the problem is in MozRepl::Plugin::JSON2, firefox 8 or MozRepl::RemoteObject.

Edit: I almost forgot to suggest Write a bug report. Since the author doesn't mention his rt.cpan queue in the documentation, I'd suggest creating a new issue in the github issue tracker for the repository.

The next option is to write a nice note to the author and inquire as to the status of Firefox8 support.

Dear $authorname,

I am a fan of your module $module. I use it to solve $my_use_case.

I see in the Changes file that Firefox8 is not listed as a currently tested browser. Is FF8 on the roadmap? How can I help test and move this feature forward?

Thank you again for your awesome module.

sincerely,
$yourname

You could also disable that safety check and see if it runs otherwise and make sure to use all 7-bit clean ascii. If you go this route, you'll be uncharted territory and responsible if things Just Don't Work(tm).

Have fun!