I can’t remember the rule now (probably to do with subjunctive or conjunctive moods but don’t quote me on that!) but I know this is correct.

The first is more like a part of a sentence in which you are complaining that he should get going on his furniture purchase. It is almost like it should end with “..because I am tired of sitting on the floor when I visit.” Even with that additional phrase the word “buys” sticks in my throat.

Present tense: 1. It’s about time he buys some furniture

Past tense 2. It’s about time he bought some furniture.

You don’t need the helping verb “that” in either case.

#2 sounds the most correct because it’s how most people say it, but that doesn’t make it correct. I keep replaying my 9th grade English lesson over and over in my head: “I wish I WERE an Oscar Meyer wiener, not I wish I WAS an Oscar Meyer wiener.”

This is correct, because you are mixing your tenses if you say “It’s about time that he buys some furniture.”