Is this feasible, or maybe just a dumb idea?

I have an old Trek 720 Multitrack from 1990 or so. It's basically a steel mixte. It was off the road and on a trainer for a few years. Last year I took it to the LBS and he put some new, but basic, components on it to see if I still liked it. Well, it's still a fun bike. I have another straight bar bike with Tiagra components and I'm thinking about swapping everything out and putting the nice components on the old bike. Would this be a waste of money? Is it even possible? I love the idea of having a good old solid bike.
My other idea is to buy a Jamis Coda Sport and maybe still swap out the components. And then I'd have yet another bike.

Don't know if it's possible - but DH uses a 730 from 1997 as his daily commuter. Those are awesome frames and I think it would still serve you well!

compatibility will probably be your biggest issue.

I think it depends on a few things. What's the purpose of the bike? How nice is the frame you're upgrading with the "nice components"? Can you get a new/used bike with a comparable frame and components?

I have a 1991 Giant ATX mountain bike with original high-end (for the time) Deore DX components, and it's still a great bike for riding short distances (running errands, farmers market and shopping runs, picnics, etc.). However, it's a pretty heavy bike with a rack (30+ lbs.), and I can definitely feel that and the limited hand positions on the straight bar when I ride it for 10 or more miles. For distances of 10+ miles, I'd definitely pick one of my road bikes, but this Giant mountain bike has sentimental value since it was my first real bike. I'm sure newer hybrid bikes (e.g. Jamis Coda Comp) would be lighter (~26-30 lbs.) but probably not by a lot, and the latest models have lower-end components (Acera, Alivio, etc. compared to Deore DX).

Tiagra components are road bike components while Deore, XT, XTR are higher-end mountain bike components that are often used on hybrids with lower gears for climbing. Tiagra isn't necessarily bad; it depends on how you ride. What were the "nice components" you had put on your Trek 720? Did your LBS replace the whole groupset (crankset, shifters, derailleurs, etc.)? How much does your LBS charge to change out the components on a bike? My LBS charges about $200-250 for that, so replacing the groupset would be $200-250 + cost of the groupset (usually starting at $500). For $750+ you could definitely get a really nice hybrid bike with high-end components (e.g. Jamis Coda Elite with Deore RD and disc brakes). However, if you have a really nice frame (e.g. rare and/or super-light lugged steel frame, titanium frame, etc.), then the upgrade would totally be worth it if the current components aren't meeting your needs.

What's the used bike market like in your area? Here in Seattle you can often find barely used "vintage" (1990s or earlier) Cromoly mountain bikes with higher-end components (Trek 930 or higher, Specialized Stumpjumpers, Giant ATX 760/770/780, higher-end Bianchi lugged steel bikes, etc.) starting around $200-300. You can also find used Jamis Coda Comps/Elites with Deore components for $250-400. These older bikes may need a tune-up, new tires/tubes, etc., but you can usually find a great bike with high-end components that will cost you less than $400-500 after tuneup. You just have to check craigslist constantly and wait for one in your size to show up.