In a run, the first save ring should be skipped either by fluttering under, or by performing a first-frame jump on the yellow platform, which launches Yoshi over the save ring. This can also mess around with your timing so be careful when jumping on these 3 platforms, run on them for a brief amount of time before jumping to avoid this problem.
After going up the first pipe, a pipe glitch can be performed, warping Yoshi into the adjacent room. This is faster than taking the arrow lift.
In a single segment run, this level should be entered with as many eggs as possible, as egging the key pot is twice as fast as pushing it.
Fast cloud door shot: 
Scrolling the screen up after pushing the boss off of the platform saves 15 frames.  It is even faster (by about 30 frames) to hit Yoshi's head on the ceiling below the level on the same frame the explosion starts, as this causes Yoshi to not jump into the keyhole for some reason.  The easiest way to do this is shown here. When you jump into Roger during the fight, press B just before landing and hold it for the rest of the fight. As Roger is falling off the ledge, hold down and then press and hold left just a bit before the pot goes offscreen. (The exact timing can depend on your TV since some TVs have some pixels cut off at the bottom). When you walk off the ledge Yoshi will automatically flutter. If you pressed left on the correct frame, Yoshi will bonk the ceiling at the right time. Note that there are a few frames of the bonk animation, so it can look like you hit on the right frame but it may still be a bit off and the fast keyhole won't work.
Boss push fast strat comparison: