Flash Player for Linux has serious issues with all non-Roman character sets.
Potlatch has a workaround for this (AFAIK, the only one in existence) but it may not be 100% foolproof. You could try upgrading to Flash Player 10.1 which is believed to have fixed the issue.
You should also make sure that your Georgian font is visible to Flash Player. I can’t provide any guidance on this, I’m afraid, because I don’t use Linux. But you should be able to find help elsewhere on the web.
Editors which run as standalone apps rather than RIAs (e.g. Merkaartor and JOSM) won’t be affected by this.
I use Flash Player on Linux and there is no problems with displaying of Russian and Georgian text. There were some problems with non-Romanian input in earlier versions of Flash Player, but current version(10.0 r45) works quite well. But there were no problems with displaying.