The day of the Lord will come as a thief in the night, and so catches many people off guard.

It says that in that day of the Lord, the heavens will pass away and the earth will be burned up.

II Peter 3:10 But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up.

 12 Looking for and hasting unto the coming of the day of God, wherein the heavens being on fire shall be dissolved, and the elements shall melt with fervent heat?

I've been taught the day of the Lord lasts for the entire thousand years of the kingdom.

That is apparently understood by:

II Peter 3:8 But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day.

That also, the burning up takes place at the end of the thousand years, still within the day of the Lord, given the interpretation above.

However, the burning of the earth would not come upon people like a thief in the night, only the beginning of the day would.

That explanation seems an over-stretch, at best.

Will the burning happen in that day, or only after 365,000 days?

Also, how are we to be "hasting unto the coming of the day of God"?