I think you mean "Oh what" (F) "fun" (Fminor) "It is to ride" (C).
In practice if I was accompanying 'Jingle Bells' in a jazz style I would probably play F7 Bb7 C (or Em7 followed by A7) creating a 'Back door' II V I, but if I was soloing over it I would think 'minor II V I in C' giving a rare opportunity to use G locrian mode over the F minor . . .
This is a great explanation. I wouldn't call this a modulation because the song isn't changing key. Rather, it's an altered chord (meaning a chord that uses notes not present in the current key), and it's borrowed from the relative minor. This works well because the two keys are closely related, but the reason the IV is changed to a iv so frequently is probably for another reason...
Generally, a chord borrowed from the relative minor will retain its function. So in the key of C major, that F major is a IV chord, and it acts as a pre-dominant, or a dominant preparation, and tends to want to move toward the V chord before resolving back to the I. If we take the F minor from the key of C minor instead, that chord would be a ii chord, which is also a dominant preparation chord. Even though it's the ii from a different key, it still wants to resolve to the V in the key of C major. This means that even though we're changing the IV into a minor chord, it still retains the same function.
In a sense, you're extending the IV chord function. This is very similar to what happens when you move from the I to the vi. Both chords generally act as a tonic chord. I is the tonic in a major key, and vi is the tonic in the relative minor. When you play I - vi, the progression doesn't really move forward the way it would if you played I - IV - V. By moving from I - vi, you're extending the I chord without playing the same chord for another bar. That's basically the same thing that happens when you move from IV to iv. You're just extending the function of the chord without playing the same chord for another bar.