To lower the volume of the game's background music, press ZR or the ZL button on the controller once. To mute the game's background music, press the ZR or the ZL button twice. To return the back ground music, press ZR or ZL button 3 times.
The ZL and ZR buttons referred here are part of the Classic Controller Pro.
It is possible to gain a boost of spped whilst driving. To do this, you must drive closely behind another kart for a certain amount of time. Once you have tailed it for long enough, wings of air will appear at each side of your kart and you will drive faster for a few seconds. This technique is supposedly known as the 'butterfly'.
In the Donkey Kong's Jungle Parkway level, you can use a Mushroom or a Star to get up the cliff in the cave not far from the finish line. This can cut seconds off your time.
In the Wario's Stadium level, it is possible to use Mushrooms/Stars at certain points to be able to jump over the walls and onto further sections off the track. Using these can cut minutes off your total time, which can help as this is a long track.
I will submit others I find at a later date, when I find them.