Based on my meditative practice I would say sitting upright would be best, and that it does make a difference. Not a huge difference, but a difference. However, I wouldn't try to never move throughout a chess tournament, or even a reasonably long game. Your muscles will start to protest and the pain will distract you. Periodically moving around, maybe even walking around, to loosen up the muscles is a good idea. Also, for best stability in a chair you want your feet flat, your lap parallel to the floor, and your knees at a right angle. Depending on your leg length, typical chairs may or may not work for that posture. I bring a cushion to sit on because my legs are long. A short person might want something to put under their feet.
Posture

People with great posture seem to have developed it at a young age, and now it's just comfortable for them. I play online chess, a lot of times, laying down. Kasparov is really interesting to watch, as he seems to not only have all these variations going on in his head, but also an audience! He played like everyone was watching him alone. He had this energy that shrouds him. Here is an fasinating video of him and Karpov that I happily recommend.
Kasparov vs Karpov in Lyon 1990
http://video.google.com/videoplay?docid=-123587302886162278&hl=en#
Great analysis, and different postures!
Once I lost to a player who crossed his legs, leaned back, with his right hands scratching his chin...



Go ahead and give it a try; slouch back in your chair, put your phone up in the air and play like that. If you’re able to get a rematch, sit up and see how you fare against the same player.
The bunny has long recovered from his limp and that's when his game went downhill.
https://www.chess.com/forum/view/community/seeking-players-who-have-a-temporary-limp-in-their-left-leg-after-stubbing-their-toe
Hi all, Im new here and really enjoying this site and many games I've had, so bravo to chess.com! Im an experienced player (I've been playing chess >40 years!) but not very accomplished, playing mostly for fun and only having played in a few tournaments in my youth. Thing I love about chess is getting into the 'zone', where I'm physically relaxed but mentally hyper. I lose all awareness of anything around me and get totally absorbed with the board.
But... recently I'm getting some kind of chess'itis, keep making silly mistakes and realised my mind is wandering because Im getting a bit fidgety, keep adjusting my seating position, leaning forward, then back, crossing legs under the chair etc. Im sure this physical unease is causing me some loss of concentration.
I tend to sit leaning forward with elbows on the table. But I noticed a friend of mine sits bolt upright, keeping his arms low on his lap while thinking, with feet flat on the floor and never changes from that position.
Is there a best posture? does it make a difference?