I've been running a Dolomite 1850HL as my daily car over winter (mostly), doing about 200 miles a week with occasional 100+ miles runs to Aberdeen and back. It's only let me down twice in the last 5000+ miles, once because I ran out of fuel and once because the thermostat jammed shut and it overheated. I also have a Dolly 1300 as well, when you get one others seem to find you...
The key will be getting it in good running order to start with. Especially if it's only been used during summer or as a show car previously, the sudden jump in mileage will cause things that were just hanging on to life to expire fairly quickly. My 1850HL was the previous owner's daily car as well so it was already "broken in" so to speak and never gave me any problems, my 1300 was another story...
Within the first 800 miles (it had only been doing about this much per year for a looooooong time and I clocked that up in about 3 months!) the engine disintegrated due to oil changes being skipped in the past, the ignition system failed, it started throwing oil out the exhaust, the starter motor gave up on life and the clutch hydraulics burst a seal.
DO away with the original ignition system and invest in an electronic one, easy to fit and you no longer have to worry about points. My 1300 had a brand new ignition system of the original design and it was crap, fitting electronic and it's never run better. You'll need to keep an eye on oil and coolant levels as well, they can drop very quickly and catch you out if you're not careful!
DIY repairs are easy on a OHV (1300 and 1500) engine and do-able on a Slant-4 (1850 and Sprint), I've stripped my 1300 engine down and taken off the cylinder head with zero prior mechanical knowledge with just the factory repair manual and this forum for help and managed to put it back together again and get it working with a basic set of spanners, socket set, screwdrivers and a torque wrench!
The "rockers" or rocker arms are a part of the top end of the engine which controls the the opening/closing of the exhaust/intake valves:
As you can see on one end of the arm is the pushrod going down to the camshaft, on the other is the valve itself. There has to be a very small gap between the end of the rocker arm and the valve spring that it presses down on, this gap will often work loose and cause a fast paced rattling sound while the engine is running. Although I have to say even the nicest Triumph OHV engines sound a bit "tappety".
If you skimp on your oil changes (or somebody else has already) your rocker arms might end up like mine did... Guess which is a new one!
Essentially though the engines are pretty robust, I've overheated mine and didn't kill it and it's clocked up 4,000 miles over the last couple of years despite having a severely worn engine. The thing drinks oil like there is no tomorrow! Keep an eye on the exhaust while the car is running, make sure there isn't any blue smoke, could indicate worn piston rings or valve seals. Bubbles appearing in the coolant bottle under the bonnet indicate imminent head gasket failure but don't worry too much about steam from the exhaust, especially on a cold start.
Listen out for a rumble when the car is first started, this is the big end bearings at the bottom of the engine, they rumble at start up due to a lack of oil to lubricate them, an unfortunate quirk of the engine. If this sound reappears while the car is on the move run away unless you want to take the whole engine out to replace them!
Given the lack of soundproofing problems will make themselves heard or felt in good time as tinweevil said. Knocks, rattles and clunks will make themselves apparent far earlier than in a modern car where the first sign of something going wrong is often it completely failing. Do keep in mind thought Dolomites are low in weight and have skinny tyres, be prepared for loss of traction on rough roads and under spirited cornering. Also get ready to spend your life doing 55mph at 3,500rpm in top gear, not sure if it's just because my example is rubbish but it'll struggle to crack 40mph on an incline and pushing the old girl over 60mph just feels a bit cruel. I did get 80mph out of her once on a downhill stretch though, bloody scary!