If a singer takes a long break from singing, be it many weeks, months, or years, how severely will it affect his ability to sing (vocal range, singing technique, etc)?
The average singer who goes from practicing or exercising many times per week down to once a week or less will see a marked decrease in their range.
Humans have a natural vocal range, or tessitura. The average is about an octave, though many can sing a span of an octave and a half or even a two-octave span (three-octave range) even "cold". Outside that range is largely a "use it or lose it" area, and the further from that range, the more effort (in the practice and technique sense, not outright "pushing") is required to maintain access to those notes.
However, this should not be permanent. If you quit singing technically for a year, then start again, you will start at your natural tessitura, which you retain simply by speaking regularly and singing along with the radio. But, as you continue to practice, your range will increase.
One last point: tessituras can shift, both naturally through aging and artificially through singing to one side or the other of your tessitura. This is not range-building; the muscles in your pharynx simply tighten and loosen in reaction to how you are using them, like cranking the tuning peg on a violin string. You don't gain any more notes, you just shift registers. Range-building requires singing toward both sides of the tessitura, which actually develops the muscles and your control over them to allow you to tighten and loosen them in a wider overall range. Good vocal coaches and performers know this, and incorporate it into their practices through extensive warmup exercises and a variation in part ranges through the repertoire.
In short, no or at least not permanently. Singing is more about learning how to relax the vocal chords and use all of your air. Basically, once you have that technique down it won't ever be taken from you (in my opinion). I believe it's like riding a bike; if you haven't ridden a bike for a while, you might suck at first. However, with seemingly minimal effort you will rapidly pick up the technique again. Same goes for singing; you might have some trouble hitting high notes and doing runs at first, but you'll start to remember all the nuances and it will come back to you in no time.
On a side note, I would not recommend abstaining from singing. There is absolutely no point. Even if you can only sing for 10 mins at time, do it.
I was not a professional singer, but I was a singer. My dad was the high school chorus teacher for 26 years. I sang. Then I became a high school teacher myself. I abused my voice just projecting it in large art classrooms. I lost my voice over about 25 years. My range is very much shorter and about an octave lower. I have since retired from teaching. My voice is very slowly showing some improvement, but it has certainly not resembled riding a bicycle after many years without one!
It's about muscle memory. All that training to feel the correct support in the diaphragm, the correct mouth and palate shape will eventually fade and need to get rebuilt. This is true for any instrument, guitar, piano, sax, etc. Your singing voice is the result of you having developed all the correct mechanics for breathing, supporting, and maintaining resonance in your cavity. If you take a short break you light hear a slight decline in vocal quality that is easy to get back with a little practice (everyone needs a break). But if you took years off I would expect that you would have lost the muscle memory to do these thing well. You might still be able to sing in tune, and sing with some power but all the old glitches you worked out through proper training would creep back. The degree to which this happens would also depend on how many years you had trained prior to stopping. I knew an accomplished opera singer who had retired. Decades later she could still project a few lines of a song (in front of friends and family onlu) with a voice so powerful and beautiful any novice would be amazed, but in her mind it was "nothing". I personally have worked with a classical voice coach for about three years and if I take weeks off when I'm on vacation the quality and control decline noticeably. These are all perishable skills.
Another factor is age. As we age we change physically and we might lose range, or our range shifts. All humans go through changes every 7-8 years depending on gender. If you took a long hiatus and during that time went through a change you might find when you pick it up again your voice is permanently different.