Can I make Do Not Disturb on my Mac active 24/7 and still check notifications manually?
Technically there's no default way to leave it on 24/7, but you can basically achieve the same thing.

Go to System Preferences > Notifications and set a schedule so that it starts one minute after it ends. For example, if you have it set to start at 7am and stop at 6:59am, it'll effectively be on for 23h 59m.

Now, because any notifications will appear for one minute, the trick is to pick a start and end time that's not when you're likely to be using your Mac. In that way, you're not going to be seeing that one minute of notifications! And, because you’ve set a schedule, it won’t turn off at midnight like it normally would.

See macOS User Guide: Use Do Not Disturb on your Mac on Apple's website.

You could also use the Energy Saver preference pane to set a schedule for when your Mac is awake/asleep and time your Do Not Disturb schedule to suit this. 

For example, Do Not Disturb can be set to be active while the display is sleeping, so if your Mac is usually on 24/7 but asleep for most of the night, you can set your Do Not Disturb schedule so that it starts at 5am and ends at 1am. While you're awake Do Not Disturb is always on, and while you're not (but your Mac is sleeping) it's still on because its display is asleep.