It was the French Impressionists who made the varnishing of paintings optional. Some of them liked the flat, matte look of dried oil paint. Today, I personally know one high-end painter who doesn't varnish his paintings.
The problem with not varnishing paintings is twofold. First, not all colors dry to a matte finish. The amount of gloss depends on oil content, how much thinner or medium was used, and also on the absorbency of the ground. An uneven gloss can detract from the beauty of an otherwise skillful painting. Second, a painting without varnish is much more difficult to clean. Paintings in the home are routinely exposed to smoke, airborne kitchen grease, pollen and other particulates that will dull and darken color over many years. With varnish, it's a simple matter of the conservator using some solvent and a light touch to remove the varnish and the accumulated grime.
All that said, painters look at varnishing as an unwelcome task. Mostly, it has to do with timing. An oil painting needs to dry for at least six months before applying a final varnish. Even though the painting might feel dry, oil dries not by evaporation but by oxidation. It doesn't "dry" so much as it "cures." Thicker passages of paint especially may still be soft and contain unpolymerized oil. A final varnish will seal the paint, keeping away the oxygen that the paint so desperately needs for curing.
But there's an easy way out - retouch varnish. I know many painters who apply a coat of retouch varnish but never bother with a final varnish. Retouch varnish can be applied at any point, since it's often used to revitalize dried, dull color during the painting process allowing for colors to be more accurately matched. When I'm done with a painting, I usually give it a couple of weeks and then apply a light coat of retouch varnish. The retouch varnish brings dull color back to life and also evens out the gloss. Typically, this is all the painting needs. Sometimes retouch varnish will seem to disappear over time, making parts of the painting look uneven again. But I don't just spray on more retouch; retouch is just a dilute version of final varnish, and multiple coats of it will seal the painting just like the final varnish. If the painting has sold, I offer to take the painting back from the buyer and varnish it.
I use either a spray varnish or a bottle with a brush. Spray varnishes tend to get clogged nozzles, leading to varnish blobs. I always do a test spray first to make sure I know how the nozzle will work. I prefer a brush, but sometimes while leaning over the painting, I end up getting a dog hair or two in the varnish. If I'm quick, I can pick them out without damaging the varnish. When I'm done, I always make a note on the back of the painting with the type of varnish used and the date. As a side note, make sure you varnish on days of low humidity. I don't know about acrylic or resin varnishes, but damar will "fog" if applied on a wet day.
By the way, if you like a matte finish to your paintings, you can get a final varnish that gives a matte finish. You can have the flat, dull look of dried oil paint and still have the protection!
Above are some of the varnishes I have in my studio today.