As a Russian I would say that this is true.
There's still a problem with any kind of paper proofs, because Chernobyl was a top-secret place and when the diaster happened, the government even tried to hide these news.
So, there are a lot of proofs from people who WERE there, but I don't think it's easy to find documents in open access.
The official version says that it really happened, and firefighters who understood that they won't survive took the place of other workers. Some of those people survived and got Order of courage, so this is actually the only proof anyone will need. I would surely say that it's not just a version, but the only existing opinion in Russia, because this disaster happened in the eyes of many people.
Important note (links are in Russian, because as I checked they show deeper info than English sources. Anyway, Google translate might help )
There're 2 opposite opinions on people's motivation to re-volunteer.
- Actually, there's information from some survivors who claim that they
were not informed about the radiation level (article from big russian
newspaper). This seems
possible, because the government didn't do well on the first stages
of the disaster - they didn't immediately inform people or started
evacuation and seemed interested in just hiding it. So, probably some
licvidators re-volunteered, because they weren't informed that they
wouldn't survive such radiation doses.
- On the other hand, there were people who were well-informed and
stayed because they thought if was right thing to do. This link
is about the Major firefighter, who stayed in the center of the disaster. There're may memories of firefighters, who share the same idea
So, we have 2 quite opposite versions. Anyway, both sources prove that people DID stay in the disaster zone. You can decide which motivation seems more reliable fro you, but some of the survivors are still alive and say that the second version is true. So, the second version of events is mostly supported by the public opinion.