[RPG] Would a monk with the Tavern Brawler feat have improvised weapons count as monk weapons

The answer to your question is No, using versatile weapons with two hands does not disqualify them as monk weapons.

But not because of any reason that made you ask the question, because your whole question is wrong as it's clear from your question you haven't understood the paragraph in question.

The paragraph in question on pg. 78, PHB says (emphasis mine):

At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are shortswords and any simple melee weapons that don't have the two-handed or heavy property.

Let's look at what that paragraph really says.

The first thing it does is stating that at 1st level a monk has a master of different combat styles using unarmed strikes and monk weapons.

The second thing it does is stating that monk weapons are shortswords and any simple melee weapon that doesn't have either the heavy property or the two-handed property.

That's everything that paragraph states. It simply tells you that monks fight unarmed or with monk weapons and then defines monk weapons as either shortswords or any simple melee weapon that isn't heavy or requires two-hands to use.

Simple melee weapon with heavy or two-handed property = not monk weapons.

Simple melee weapon without heavy or two-handed property = monk weapons.

So what about the versatile property? A weapon with that property just gives you the option to use two hands during an attack for greater damage. However, in this case whether a weapon has the versatile property or not has no impact on the RAW as it only purpose in this case is to define what a monk weapon is, which is a shortsword or any simple melee weapon without either the heavy property or the two-handed property. Any other properties such as light, finesse, reach or versatile has in this case no bearing on the rules. Nor does how you use said weapon.

Monk: Yes.

Monks can choose to use their attack's original damage or to use their Martial Arts damage dice. This often comes up since at lower levels, many monk weapons do higher damage than the Martial Arts dice can. As your quoted section says, monks can roll the Martial Arts damage dice instead of the normal damage.

Tavern Brawler: No.

Tavern brawler outright changes your damage dice for unarmed strike to 1d4. It doesn't have the language that Martial Arts does of allowing you to choose which damage amount to use.