Nope not really. I mean if you care about uniforms, Shatner's weight and makeup changes, sure it makes a difference at the start. Spock also develops as Nimoy and the writers got a handle on him. From The Managerie on? Nope.
In fact, for the most part, it wasn't really supposed to be seen in production order. The Cage and Where No Man Has Gone Before weren't intended to be seen at all, but production delays made it necessary to use them. From what I've read, The Corbomite Maneuver was the preferred episode to start, followed by The Man Trap and then episodes would be dropped where most effective. As others have said, it wasn't a serial, so nobody really cared where an episode landed.
For example, Catspaw was the first episode shot for the second season, but they fully intended to hold it back for Halloween and kick off the season with a strong Spock episode: Amok Time. Production order is based as much upon scheduling and necessity as airdate. Grouping episodes with similar sets and locations, guest star scheduling, or simply having the script ready. City on the Edge of Forever was so late, it was dropped at the end of the season - which back then was a graveyard for weaker episodes because audiences had fallen off by then. This was well before big Season Finales drew audiences in. A top class episode like this should have been seen in sweeps periods.
So no, for episodic TV like the original Star Trek, it only makes a superficial difference. How you prefer it is all personal. I don't care either way, since I rarely watch the series from start to finish. I usually cherry pick.