In previous Mario Kart games, coins collected during races have increased the player's speed. The primary purpose of coins in Mario Kart Tour seems to be to increase score and to spend later, but do they increase speed as well?
No they do not. (in the way that you are thinking)
I tested this out, and it appear that coins do not give an overall increase of max speed in Mario Kart Tour like coins have in previous games. However, if you look closely, at the moment you collect a coin little streaks of air appear around the screen, seemingly indicating that there is a small temporary boost in speed right when you pick the coin up.
I recorded two similar races side-by-side, and you can watch them below. In one video I tried to avoid coins as much as possible (I think I collected 5 throughout the race) and in the other I collect as many coins as I can (I think I end up with 24). You can watch when I pass a tuft of grass, or any other object in the environment, and count the seconds between when I pass another marker in the environment. The amount of time is the same regardless of if I have many or few coins.