Is it a residential building requirement to caulk between a bathroom tile floor and a wood base?
No, the code does not specifically require caulking between anything in a bathroom. However, it does require, “a non-absorbent floor and walls up a minimum of 6’ for tubs and showers”. (See IRC R307.2)

However, as we all know, that wood you’re concerned about will absorb moisture on the “end-grain” very easily. You should protect it with a few coats of sealer or paint to protect it...especially in a moist environment like your bathroom.