Is there any chance that someone could sue them for filing illegitimate claims?
To prove tortious interference with a contract, a plaintiff must establish several elements, including that The defendant’s conduct led to a breach of the contract; and The plaintiff suffered damage as a result.
The question would be whether Github users suffered a breach of contract for having any of our repos potentially shut down for a reason that involves a third party's claims of violation of DMCA.
There's also unseen elements at play. If you're publishing semi-anonymously, the counter-claim process can deanonymize you.
Republishing the same content at a slightly different URL is the obvious, cheapest solution.
Very few people have been able to claim damages from false DMCA takedowns, although the DMCA specifically allows for it.