I'm also very interested to understand the reason of such problems.
1. If DBeaver has a chance to start then all log files are stored in USER_HOME/.dbeaver/.metadata directory. There should be a set of *.log files.
2. If something very horrible happened and osgi framework was unable to even start then some log files may be stored in DBEAVER_INSTALL_DIR/configuration folder.
I believe that your case is number 1 (at least you see DBEaver logo - it means DBEaver started to load it's Java components).
PS Eclipse plugin is not working in current version. We are going to make it fully-functional in next major version (DBeaver 2.0)