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, but this is hard to time. 
Boss push fast strat comparison: