Correct me if im wrong - Cyclebutane is puckered because all the carbons are SP3 so they want to be 109.5 bond angles but can't be because its arranged in a square, which we know has 90 degree angles on every side. Oxetane is virtually flat because the oxygen has two lone pairs rather than hydrogens so it can flex a bit more to flatten out the ring. And actually the electron groups can push the bond angle to be smaller like they do in H2O which has an actual bond angle of 104.5.
Edit: it would be easier to understand if you made the molecules with a model kit
Puckering will slightly increase the internal bond angles, and cause added strain. However, this is more than offset by the relief of steric strain from interactions between neighboring hydrogen atoms, which the planar model imposes. Oxetane which has an oxygen in the ring, or even thietane which has a sulfur in the ring have been shown to be planar. There is no need to pucker. Evidently, when a heteroatom replaces a methylene group...i.e CH2 group, it REDUCES the number of steric interactions. Since steric interactions are reduced, there is little need to change to the puckered conformation. I hope this helps on a challenging Organic Chemistry problem ! I give you a gold star !!!