Are tracks dynamic in Forza Horizon 3?

Handling is definitely affected by weather conditions. The easiest way to see this is to build a 'grip-roll' car, with something like the Ford Woody. Highest grip tyres you can get, rally suspension, then loosen it off all the way. When it's dry, the car is very loose handling but unlikely to roll on a straight. In the rain, the car will roll with the slightest provocation.

Puddles you see form on the road under heavy rain do not affect handling in any special way (for example, you won't aquaplane on them). The only bodies of water that affect handling are the ones that exist in dry conditions as well.

Forza Horizon is much more lenient and "arcadey" than its Motorsport Counterpart, Puddles don't affect your handling in a drastic way like in Motorsport 6, but the different road condition will still affect your handling and braking. Unfortunately I don't have anyway to prove these claims on hand, but something you can do is press down on the D-Pad to bring up car stats, and cycle through the screens with left and right on the D-Pad to compare differences in wet and dry conditions (tire grip/friction will probably be the best screen to check, tire temp as well (wet roads could help keep them cooler)