My EEVBlog BM235 just started doing the same thing as well: it starts up, but gets stuck on the LCD test screen and is completely unresponsive. If I hold down the HOLD key at startup, mine shows 9, which I think is just the firmware version.
I tried fresh batteries, checked fuses, took it apart, did a visual inspection of the board, cleaned the wipers & contacts, and scoured google for solutions. This thread seems is the only place where I found the same issue mentioned, so adding my observations:
I found a highly impractical workaround, at least for my case: I noticed is that if I hooked it up to a DC power supply, and LOWERED the voltage to about 2.1 volts, it started up normally (albeit with a low battery warning and a dim screen)... Turning the voltage back up to 3V got it behaving properly as far as I can tell, but the problem returned after powering it off. I could also get it to start up properly if I very briefly disconnected the battery (probably because that briefly dropped the voltage via some partially-discharged capacitor)... so maybe the problem has to do with some transient voltage during startup(?) - maybe a capacitor that' s not charging? I'm a mech eng, so out of my depth with this stuff, but maybe that can help someone smarter than me to properly diagnose the problem...