If he has had some teeth removed, he may have food stuck in between the teeth and/or the gum. My elderly girl is the only one of my Bs who does this and she has had teeth removed. This happens a couple times a week and I use my finger to help loosen and remove the food.
Shaye is 3, and she also does this - she is kind of a messy eater, carrying food away from the bowl and dropping pieces along the way to be picked up on her way back to the bowl, and she inevitably gets food around her mouth while digging into it. When she's done eating she "washes" both sides of her face with her paws but then comes and wipes it further on us, the furniture, or whatever suits her fancy, so it is likely he's using the couch as a napkin since he's older and it's less effort to do it that way. But, checking his teeth is a good precautionary move, and maybe helping to loosen stuff if he has missing teeth.
Maybe try brushing his teeth after he eats? I agree, if this is a relatively new habit, it may be teeth related…..but I am not clear about whether the face rubbing in the couch is new, or just the Christmas tree rubbing? If he has always rubbed his face in the couch, I would just say typical Basenji.
Becca is about 6 years old and has always done this. In the morning when she gets up she stretches then rubs her face on the carpet to rub the sleep out of her eyes. Then after she is finished eating (especially something really tasty) she will run to the rug and rub her face again then paw at her cheeks and lick her paws to make sure she doesn't have any crummies on her face.
I have had some who do it, some who don't. Always good to check the teeth. In fact, a good idea to check and to start brushing their teeth regularly when young. I find it alarming that nearly half of all dogs have dental issues by about 3 yrs of age. My dogs always had perfect teeth til they died, but I used to feed RMB. Now, ugh. Fortunately Cara is a nut job who actually LOVES to have hers brushed. Arwen has learned to tolerate it.
I can honestly say that I was NOT the greatest at teeth with my first Basenjis… and I can tell you that I learned a hard lesson on how important it is not only to brush (and I swear by Placque Off but I still brush at least 5 times a week) but to make sure that the teeth are checked and if there is any question a closer check is needed...