Besides avoiding mistakes, a big way to save time throughout the run is by minimizing airtime. You move at 6500 speed on the ground and 5200 speed in the air (25% slower), so you lose frames for unnecessary time spent in the air.

Jump Cancelling

When you hold B, you start flying which lets you stay in the air longer and reach slightly higher places. However, if you just tap B, you can use it to kill your upward momentum. Jump cancelling saves time across a run because you can do it pretty much whenever you jump or kill enemies to make sure you aren't in the air longer than you need to be.

Do be mindful of your energy: tapping B will use a little energy unless B is held for only one frame. This should only matter if you're playing super optimally like for an IL, you can just not do it in levels where you need lots of energy.

Damage Boosting

When you take damage, you get knocked into the air in the direction of the side you hit the hazard on. You move at 6500 speed while you have the hurt sprite and quickly return to 5200 speed when it ends (assuming you're still in the air). While invulnerable after getting hit, you can walk through other hazards without stopping. There are various places in the run where intentional damage boosting can save a few seconds over playing normally. The main thing to be careful of is to make sure you hit the side of the hazard in the direction you're moving so you don't get knocked backwards (i.e. if you're going right, you want to be hit on the left).

Ledge Clipping

When you stop flying at the right height next to a ledge, you can clip slightly into the floor. There are two spots in the run where it can save time by getting you low enough to avoid falling spikes, but they are precise enough to probably be TAS-only (21:54 and 33:46 in the TAS below).

Mushroom Cancelling

TAS-only tech unless you're insane. If you cancel a mushroom bounce on the first possible frame with a one-frame B tap, you land on the mushroom and can walk on it without bouncing. This is done a few times in the TAS but isn't realistic RTA.

Miscellaneous Info

You bob up and down while flying. Timing the start of short flights to avoid bobbing can let you reach your destination quicker.

You move at 9600 speed (~47% faster) walking on a platform moving in the same direction and 2600 speed (40% slower) walking against it.

Hitboxes in this game seem to be roughly the shape of their sprite. What this means is landing on the top of an enemy's head will start your jump higher than if you landed lower on the side of them. This also applies to things like flowers and mushrooms.

Notable Skips

2-4 Skip: 13:35 in the TAS below. You can jump on the flying bug then fly up to the higher section instead of going the long way around. If you kill the bug but don't reach the high platform, you can still just fly to the ledge below it from one of the thin pillars.

4-5 Skip: 29:27 in the TAS below. You can go right at the start, drop down and time your flight to land on the right ledge, then follow the TAS route from there. You can also hold left to land on that ledge and jump across from there which is less risky. You do lose time if you fall all the way down.

5-4 Skip: 35:25 in the TAS below. Just hold left, hold A once you're in the air, and start flying when you reach the peak of your jump off the enemy and you can reach the left platforms. This is the largest skip in the game and probably saves around 40 seconds. The platform cycle right after is very hard though and requires tight jumping to make it, if you do it's another several seconds.