I'm not terribly familiar with the nuances of kombucha, but I suspect the answer lies between "maybe" and "probably". The key question is, how similar the finished kombucha tastes like the original tea used - the more that is the case, the more the initial tea quality will matter. The few times I've had kombucha, it has seemed not terribly similar to plain tea - thus I would think the difference between kombuchas made with different grades of teas would be subtle.
Higher quality ingredients do tend to improve the quality of the finished product, however, the more processed or mixed something is, the less any individual ingredient matters. Tea is a major ingredient in kombucha, so changing the grade should make a difference in the final taste. Kombucha is fermented and sweetened, so I suspect a fair amount of the differences between grades of tea would be smoothed down by that much processing - not all the differences would be lost, but perhaps enough to make the change subtle, rather than overt. The same would go for different types of tea, the greater the difference, the more change in the final product. It will also matter how picky you are, in terms of telling the differences between grades of tea leaves - some people find the differences very obvious, others very subtle. If the grade of tea matters to you a lot in regular tea drinking, you would be more likely to tell the difference in your kombucha than someone like me, who is fairly causal about different grades of tea and is fine with most anything.
I would guess that you can make a perfectly acceptable kombucha from regular tea bags. Also, the tea doesn't seem to be carefully brewed, just steeped in boiling hot water until all the flavor is extracted - so the differences between whole leaf, broken leaf, and powder (assuming the same grade of tea) will vanish in the brewing - a lot of the differences in processing tend to matter the most with certain methods and timing of brewing, like brewing at lower temperatures or controlling the steep times. This is more or less the same reason tea grades don't matter very much in preparations like chai tea or milk tea - strongly brewed (overbrewed, most would consider it) and doctored with other flavors, the grade of the tea itself matters much less than a precisely brewed plain cup.
Differences in the grade of tea, high vs low quality or leaf vs bud, and the bigger differences with type of tea, green or black or puer-eh or oolong, will matter a little more (as there are different compounds being extracted, or extracted in different ratios). I would expect differences in color and taste would be the result of using very different kinds of tea to make kombucha - though I don't know how they would actually come out, without experimenting.