What would happen if I put my smartphone into the refrigerator?

Nothing
It’s sort of anticlimactic, really. Your phone will sit there until you take it out.
In the freezer?
Well that’s a different story. Your phone will slow down noticeably (electronics under colder conditions slow down), and it may ruin the battery.
Also, some condensation in the freezer may seep into the phone and short out the circuits if you attempt to turn it on.
I don’t think warranty covers freezing your phone.