Short answer: most elements of the engine are designed around a nominal operating temperature, short journeys don't get the engine up to that temperature so everything operates suboptimally, causing wear which will shorten the lifespan of these elements.
Long answer: Whether diesel or petrol, engines are designed to function based on the energy they can extract from the fuel they are given. In most cases, the combustion of that fuel has a certain level at which it is optimal. For petrol, an air/fuel mixture of 14.7:1 and for diesel 14.5:1 is the stoichiometric ratio at which complete combustion of the air/fuel mixture occurs. Anything deviating from this ratio results in incomplete combustion, which produces byproducts such as soot, carbon, and nitrous oxides at levels higher than normal. Therefore, everything in the engine is designed to best achieve this air/fuel ratio.
Assuming this goal, the first thing an engine has to do when it starts combusting is to ensure it can turn over. Everything in the engine that moves needs to be lubricated, and engine oils work within a specific range. That's why we see viscosity ratings like 10-40W, 5-30 etc. Essentially, the engine oil needs to be thick enough when the engine isn't running to be able to stick to all the moving parts so that they're lubricated when it first starts turning, but also thin enough to get into every nook and cranny that needs lubrication. The thing about oil is that its properties change based on temperature, which is why older mineral oils had very limited operating ranges. Synthetic oils have overcome this issue to a great degree, and keep their characteristics both when it's quite cold out and the engine is just started, and when you've been thrashing it around a race track in the summer. Unfortunately, the limitations of physics means that oils are at their worst when the engine is cold, and only start doing their job properly once they have reached nominal operating temperature. That's where constant short journeys start to come in.
When used for short journeys alone, the engine rarely has enough time to get up to temperature (through both friction and the heat produced by combustion). Therefore, many of the engines moving parts are poorly lubricated during this period. Additionally, since engine temperatures aren't where they should be yet, combustion itself is usually poor as oil hasn't had time to get between the piston rings and cylinder wall, and the cylinder and heads/valves have differing temperatures. Since most of the engine is metal, the differing temperatures cause differing rates of expansion. That means that everything that should be sealed tight (such as the aforementioned piston rings) haven't had time to warm up and expand to provide a proper seal. All in all, the engine is basically working to both provide motion, and get up to temperature (through extra fuelling especially in the cold) and try to maintain a usable air/fuel mixture. Since during the short journey the engine is under higher load but not in the optimal range of doing that work, it wears out certain things.
The things that get worn out most due to short journeys are: oil, spark plugs, injectors, catalysts, turbos, and seals. Oil, turbos and seals (such as piston rings, valve seats, exhaust gas recirculation - EGR valves etc.) get worn due to the moving parts issue. Plugs and injectors get worn due to the thick oil and poor combustion fouling them (this also influences EGR valves and the diesel particulate filter). Catalysts degrade because they need to be at a 'just right' temperature to operate best. Too cold, and condensation corrodes them. Too hot and the catalyst gets consumed through high temperature oxidation (same corrosion basically, but due to temperature excess rather than condensation of water).
All in all, overall engine life (while hard to judge) won't be dramatically reduced from a life of short journeys, but the bits of it that take the brunt of the force of these short journeys will need to be replaced more frequently. Early (as in about 2-3k kilometers sooner than recommended) preventative maintenance would minimize the issue, with closer oil and spark plug changes working in your favor. The greatest issue would be if something like a piston ring, EGR valve, injector or diesel particulate filter going bad as they're expensive (especially if it's an internal element like the piston ring or valve seat that would require opening up the engine). Oddly enough, something that may work in your favor (especially if we're talking about a petrol vehicle) is the occasional 'Italian tune-up', or basically some spirited driving (once the engine has warmed up). It clears out carbon buildup and gets oil into all the places it hasn't had a chance to go in the short journeys. If that's also not an option, a short (20 minute) motorway stint once a month would also work in your favor.
