I removed both RAM sticks and turned it on and it does exactly the same thing.  Powers on, proc fan is on (USB KB lights are on), but no video and no beeping at all.  It runs for about a minute and then shuts off.

Hello,

remove RAM and reboot, if machine beeps, one of the RAM modules will be faulty

I wish it would show me the startup menu - no video at all on any input.  It may indeed be toast - the person in the office where it came from said they thought it had been dropped (?).  So it may be damaged - just wanted to troubleshoot it before parting it out.  Thanks for the suggestions.

Solid orange power light can mean bad MB.  Hopefully that's not the case with a somewhat newer PC.

At startup, immediately try going into F2 BIOS and/or F12 Boot Menu.  If it doesn't let you do that, the MB might be a brick.

Any recent updates?

Dropping would certainly account for the problems. @bradthetechnut is spot on with the MB and amber light. If you pull the RAM and you hear no beeps when trying to boot, it would seem to mean that the MB has no idea the RAM is missing and that could easily be a bad MB. One wild shot here though. If it was dropped could the CMOS battery be dislodged or have fallen out of the MB?

Also I see there are some $30 used replacement mobos on ebay - I may get one and replace it.  It's a very cute little system - I have never had one so small!

Hello, I thought this same thing - but no, the CMOS is there and firmly in it's slot.  Everything looks in place - nothing loose, nothing fell out of it.  But without the RAM when I plug in, the light is amber for a split second then goes white - then the machine fan runs for about 1 minute - then it just cuts off (no lights, no fan).  At no point does it beep or output video.

MB's can be quite reasonable on eBay and is a good idea.  Be sure it comes with something for a return policy.

You're welcome for the suggestions too.