The biggest symptom of any rotating mass that isn't balanced is going to be vibration. It'll be different than the vibration you feel from something like a knobby tire, which buzzes even at low speed. For an out of balance tire, you won't notice it while you're cruising around a parking lot at 15 mph, but the faster you go, the worse it would get. How bad it gets, and how long it takes until the vibration is noticeable, would depend on how out of balance the tire was.
I'm not a physicist or engineer, so I don't know how much weight it would take in an out of balance tire to create that effect. I've never used more than an ounce of weight (four 1/4 ounce wheel weights) to balance a tire. That's about the weight of a half of a hard boiled egg. I honestly don't think that small of an amount of weight (28 grams) will make a lot of difference in a tire that weighs about 17 pounds (the weight of a rear Shinko 705) and isn't spinning at 10,000 RPMs.
If you're really curious, try this easy experiment. Balance your tire and put it on the bike. Now mark where the weights are on a balanced tire, and move them to the opposite side of the tire. Now the tire is going to be out of balance by not only the weight of the heavy spot of the tire, but also by the weights themselves. Now, go ride the bike at a brisk pace, and see if that "doubling down" on the heavy side of the tire makes it vibrate. If it does, now you'll know what it would feel like, and once you know, you can move the weights back to their original balance point without having to take the tire off and rebalance it. If you don't notice anything, then you'll know that balancing the tire was actually a waste of effort in the first place.
I may try this experiment myself next time to see how necessary it is to balance a tire.
"No matter where you go, there you are."