Check 8085 reset signal and clock oscillator. Reset signals are often generated using cheap electrolytics that dry up.
Ok, the oscillator for the 8085 is running - Will take another look at the reset line, however, I am getting a green status LED on the processor board, so I think this is a good thing
Check ALE and R/W signals are active
I am a little thick in this area, Analogue stuff I know, but digital is a new thing to me.
Check that all address and data bus lines are in sensible states at the CPU and not stuck.
As above
Get some idea of the memory map. An 8085 will have ROM at address 0 upwards, I think, but other things (RAM, video and so on) could be anywhere. Address decoding usually involves a 74LS138 somewhere! Check that the signals on its pins are making sense.
Lost me completely 
Check that the ROM is getting enabled for at least a few clock cycles after reset.
Drowning now
See if anything is trying to access the RAM - do its chip select lines ever go active?
Will try to get a screenshot of the RAM pins - there is something going on there, quite what it is I do not know
Is the reset signal pulsing? If so, there might be a watchdog timer firing. Find out what should reset the watchdog timer - there will be a flipflop somewhere triggered by access to a particular address.
From memory the reset line is staying high - will report back on this
See if the processor is stuck in a tight loop. It should be possible to spot this on a scope by looking at the address bus. Looking at the state of the address lines will show where it's getting instructions from. Is that within the ROM address range?
Sorry, lost me again
Check any bus buffers (74LS245 is a popular choice) and make sure their inputs and outputs are consistent some of them are bidirectional.
The 8085 has a multiplexed address/data bus. Find the address latch (probably a 74LS374 or similar) and see if it's behaving itself.
I bow to your superior knowledge - I wouldnt know if a multilegged fuse was behaving itself or otherwise
A quick-and-dirty method of checking RAM is to simply piggyback another chip on top of each one. This isn't 100% reliable but if it makes a difference, you know you've found a problem. In the case of this board with its 1-bit RAM chips, I might be tempted to remove them all and wire in a 6116 or similar to get it going and eliminate RAM from the equation. Spare ROM sockets can be handy for this - most of the pinout is the same.
Okay, I will look out for some suitable RAM just in case
Pull out the ROM(s) and read them on an EPROM programmer. Do they look sane? Can you read any text in there? Even bizarre non-ASCII devices seem to often use ASCII text internally. Disassemble the first few bytes. Where does it jump to?
I
I have some access to an EPROM programmer, I will see if I can read the chips - otherwise a bit lost with disassembly.
These tests can give you a clue as to what's going on. Note that none of them use a logic analyser. My weapon of choice for this is a decent analogue scope, though a modern digital one that's good for troubleshooting would be even better. A bad digital scope will just be misleading.
I have a Tek 465, and a logic probe, does this qualify?
I don't know if any if this helps, but if I had the machine in front of me I'm sure we could get some life out of it!
Was this a veiled offer?