Though this thread is now almost 3 years old, I wanted to append my little bit of an HP 3325A success story to it so that then next intrepid traveller who Googles their way to this topic with their busted 3325A will have one more breadcrumb to add to their path.
I received a intermittently functional 3325A from an ebay "as is" purchase, where it would only occasionally power up into a functional state, but more often than not power up with a fan-only, or fan with a random button lit or scrambled LED display hieroglyphics. If I let the unit sit powered on for anywhere between 5 to 20 minutes, it would then typically power up successfully and mostly behave itself.
I tried being all technical about troubleshooting this thing, which among other things involved Google searching every which-way I could think of one of those which ways found me here on the EEVblog on this thread. Though my particular problem was not exactly like Lorenzo_1 or rf+tech described, it gave me some good information for thought and generally led me in the right direction.
To cut ot the chase, the offending component in my case turned out to be u14 on A6, a 74LS174.
To locate this part as the faulty one, instead of getting all super technical like Lorenzo_1 and rf+tech, and since my unit's faults seemed to be warm up related, I employed the old Freeze Spray and Hot Air station to divide and conquer my way across A6, until I singled out u14. Freeze it, and the 3325 will fail to start up; warm it up, and bingo, working like a champ.
For those of you in a similar fix with a complex machine like this, perhaps also like myself lacking in the specialized equipment and for that matter the specialized experience, good old freeze spray and heat might be your saviour too.
Best of luck.
Moo.
[Oh. And P.S. I too happened to have used a Tek 485 in my efforts. Faithful old friend of a scope, it is; long may its little screen flicker...]