DS1307 timekeeping date significance (leap seconds?)
You may very well operate the RTC chip by leaving the "date" field at its initial default value. You will find however, as you enable the RTC to count the "time" field that once each 24 hours the "date" field will accumulate another day.

Setting the "date" field or not will not be having any impact on the RTC's accuracy or in its power consumption in any realistically measurable way.

I have not worked with any RTC's with an "daylight savings time feature" (DST) but if there was an RTC with such feature and it was enabled you would see that interact with your "time" field. In that case you would probably want to have your time piece comprehend the date. Like I said though it is unlikely to find an RTC chip with DST feature as such feature is different in many parts of the world and is subject to change in any particular area at the whim of governmental types.