I leave my pfSense hardware firewall built into an old Fujitsu PC running for weeks on end. Only when I go on holiday do I shut down the system. At only 20W, it's not too costly to keep the firewall switched on 24/7.
Home broadband routers are often switched on all the time to provide internet access and WiFi, day and night.
At work, some people leave desktop computers switched on at night and don't bother turning them off at the weekend. A waste of power certainly, but computers can operate like this for years.
Servers and workstations remain powered on for months with no ill effects. When you do get failures in such systems, it's often at switch on. Computers suffer less stress when they're not switched on and off thousands of time.
If you want to leave your Minecraft computer running for a week and can afford the electricity, create an image of the C: drive with Macrium Reflect in case of problems, then turn off the video display
It might be good idea to switch on the screen each day to monitor the spawners and check if everything's OK. If the computer has crashed, you'll have to come up with a different strategy.