Rear derailleur range
 

I have been slowly swapping out old components for new and am down to the final piece - the rear derailleur. I am running an 8 speed 12-25 cassette with an old broken down Deore LX RD. I have done some reading and know that 7 speed cassettes are a little narrower than 8 speed, for obvious reasons. My question is, are current 7 speed rear derailleurs capable of the range to support a span of 8 cogs? I'd like to go Tourney to keep it on the cheap, otherwise I'm into Sora.

It should work. I've run 7-speed derailluers on 9-speed with no problems.

(Avoid pre-9-speed Dura-Ace rear derailleurs, different actuation ratio)

Yes, a 7-speed rear derailleur is fully capable of shifting over an 8-speed (or 9 or 10 for that matter) cassette. You will have to adjust the limit screws to match the width of the new cassette but otherwise no problems.

I'm currently using a 7-speed STX and an older 7-speed RSX rear derailleurs on two different 8-speed bikes with no problems. BTW, the Sora would be a worthwhile upgrade over the Tourney, which is the lowest line rd Shimano makes. The cost difference can't be much.

The rear derailler doesn't care about 7, 8, or whatever speed. There are exceptions on some groups due to cable-pull, but I doubt this applies in your case. Only thing that matters is that the cage can wrap the amount of chain you have. The 7-8 speed mechanism is in the shifter. As others pointed out, make sure you have the limit screws set so you don't throw the chain into the spokes or dropout and you should be good to go.