This would not be imbalancing and only affects a small-ish (5-29) number of specific things

Unarmed strikes already activate and qualify for various features due to them counting as "attacks", "melee attacks", "weapon attacks", and "melee weapon attacks". As shown in this Q/A the only things unarmed strikes currently cannot qualify for are attacks which require you to use weapons (also ranged attacks of any kind, but making unarmed strikes count as weapons would not change that fact).

Notably, unarmed strikes cannot be ranged and do less damage than any other weapons, this means that allowing them when a feature requires "an attack with a weapon" would not increase the versatility of these features. One situation this change would help with is if you had no weapons available (such as after being capture) you would now be able to use an unarmed strike to activate these features; the only features which this affects meaningfully are divine smites and the blade cantrips as they could be used even when completely without weapons adding a sizeable amount of damage. However scenarios where you have absolutely no weapons available are situational and few and far between so I would not call them imbalancing.
Additionally there are features which already improve unarmed strikes (such as the Monk's martial arts dice) and combining these with unarmed strikes counting as weapons would not create significant imbalance, especially since every Monk feature that allows Monk Weapons to be used also allows unarmed strike (there is one exception discussed later).
Having unarmed strikes count as weapons gives very slight buffs to features in specific situations but these are rare and small enough that they are not imbalancing.

I will now provide a full list of the effects this change would have (well, a list of all the things I could find):

The features affected

The Paladin’s Improved Divine Smite feature states:

Whenever you hit a creature with a melee weapon...

So you could use improved divine smite with an unarmed strike but this would only help you if you were weren't wielding any weapons at the time, or were facing a for you would not want to use weapons against such as those that corrode metal weapons.

Extremely technically the Crown Paladin’s Exalted Champion feature states:

You have resistance to bludgeoning, piercing, and slashing damage from nonmagical weapons...

Normally unarmed strikes are not weapons, and so they would bypass this resistance; however, this same phrase existed in the Monster Manual and a few places in the Player’s Handbook and errata have since changed instances of “nonmagical weapons” to “nonmagical attacks”. This is also shown in this Q/A.
It is likely this usage for the Crown Paladin was simply not noticed, and that it should say “nonmagical attacks”. So technically, if you read the feature incredibly strictly, counting unarmed strikes as weapons would change the feature.

The enlarge/reduce spell states:

The target's weapons also grow to match its new size. While these weapons are enlarged, the target's attacks with them deal 1d4 extra damage...

And for the reduction option the opposite occurs. Normally this change to damage only affects weapons and so not unarmed strikes, but now it would affect them as well, this change is small but would benefit Monks using Flurry of Blows and similar features.

There is also the Great Weapon Master feat which states:

When you score a critical hit with a melee weapon or reduce a creature to 0 hit points with one...

Normally this cannot apply to unarmed strikes as they are not weapons, but with the change it could.
Similarly the two blade cantrips (green-flame blade and booming blade) say this:

You must make a melee attack with a weapon...

So they would usually not work with unarmed strikes, but with this change they would, again this change only has much of an affect if you had no weapons available.

Can you touch or wield an unarmed strike?

There are some features which make having unarmed strikes count as weapons somewhat confusing:

There are five similar features: The Forge Cleric's Blessing of the Forge feature, the Hexblade Warlock’s Hex Warrior feature, the magic weapon spell, the elemental weapon spell, and the holy weapon spell. All of these have similar descriptions effectively containing the phrase:

You touch a weapon...

Assuming you can touch an unarmed strike, you would now be able to use these affects on them (what this means is not clear and is not laid out in the rules elsewhere so it would be up to the GM to determine how this change would work).

There is also the Devotion Paladin’s Channel Divinity: Sacred Weapon feature states:

You can imbue one weapon that you are holding with positive energy…

Similarly there is the Warlock’s Pact of the Blade feature which states:

You can use your action to create a pact weapon in your empty hand. You can choose the form that this melee weapon takes each time you create it…

Thus they could create unarmed strikes (what this means is up to the GM) which would gain the usual benefits of summoned weapons such as overcoming resistance and immunity to nonmagical attacks and damage.

All of the Warlock’s Eldritch Invocations which usually only work with pact weapons would now potentially also work with unarmed strikes.
These are the Lifedrinker Invocation which states:

When you hit a creature with your pact weapon...

The Thirsting Blade Invocation which states:

You can attack with your pact weapon twice...

The Eldritch Smite Invocation which states:

Once per turn when you hit a creature with your pact weapon...

And the Improved Pact Weapon Invocation which states:

You can use any weapon you summon with your Pact of the Blade feature as a spellcasting focus for your warlock spells...

Though what using an unarmed strike as a spellcasting focus would mean it up to a GM.

Finally the Dueling Fighting Style states:

When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon…

This would likely not need a ruling as you can’t really wield an unarmed strike, but it’s an interesting case nonetheless.

Remember the above section of changes is all from the confusing idea of summoning, touching, or interacting with an unarmed strike as you would a weapon.

Are unarmed strikes simple or martial weapons, or perhaps neither?

One thing you should decide is whether you would want unarmed strikes to count as simple or martial weapons, or neither. Fortunately this does not affect proficiencies gained by classes or races as the Player's Handbook Errata states:

You are proficient with your unarmed strikes...

However a decision here would impact the spell Tenser's transformation whose description states:

You have advantage on attack rolls that you make with simple and martial weapons...

Normally Tenser's transformation would not affect unarmed strikes, but it would if they were classified as either simple or martial.

It would also impact the Swords Bard's Bonus Proficiencies which states:

If you're proficient with a simple or martial melee weapon, you can use it as a spellcasting focus for your bard spells.

And so Swords Bards could use unarmed strikes as a spell casting focus (what that means would be up to a GM).

Additionally the Orcish Fury feat states:

When you hit with an attack using a simple or martial weapon you can roll one of the weapon's damage dice...

And so the feat would now be able to apply to unarmed strikes; however, unarmed strikes do not typically have damage dice as shown in this Q/A so it would only apply if you were a Monk or something else gave your unarmed strikes damage dice such as natural weapons or the Tavern Brawler feat.

And finally the Kensei Monk, which proves to be more complicated. This is because the Kensei Monk's Kensei Weapons feature which states:

Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties... Weapons of the chosen types are monk weapons for you.

This change would allow Kensei Monks to choose unarmed strikes as one of their Kensei Weapons, and from that a few of their features would work differently:
The Kensei Monk’s Deft Strike feature states:

When you hit a target with a kensei weapon...

And their Sharpen the Blade feature states:

As a bonus action, you can expend up to 3 ki points to grant one kensei weapon you touch a bonus to attack and damage rolls when you attack with it...

Both of these features would work with unarmed strikes if the Kensei Monk chose them as one of their Kensei Weapons. In addition, unarmed strikes would now count as monk weapons for the Kensei Monk which would cause two more things to occur:
Their Magical Kensei Weapons feature could apply to unarmed strikes but their unarmed strikes already count as magical from their Ki-Empowered Strikes feature so this wouldn’t create any actual change.
However their Unerring Accuracy feature would now work with unarmed strikes as the feature states:

If you miss with an attack roll using a monk weapon on your turn, you can reroll it...

Remember, the above section of changes would only occur if unarmed strikes were classified as either simple or martial weapons.

Unarmed strikes don’t have weapon damage dice

There are also six features I found which would be affected by this change but for which it is not readily apparent. These are all features which very technically do not work with unarmed strikes because they are not weapons:

A Paladin’s Divine Smite feature states:

When you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage...

As shown in this Q/A Divine Smite does not technically work with unarmed strikes, though they do qualify for melee weapon attacks there is no “weapon’s damage” to be modified as unarmed strikes are not weapons.
Similarly the Savage Attacker feat states:

When you roll damage for a melee weapon attack, you can reroll the weapon's damage dice and use either total.

And so though unarmed strikes qualify as melee weapon attacks, there are no “weapon’s damage dice” to reroll and so they do not work together. This is shown in the Sage Advice Compendium which states:

Does the Savage Attacker feat work with unarmed strikes?
No. Savage Attacker relies on a weapon’s damage dice, and an unarmed strike isn’t a weapon...

In much the same way the Half-Orc’s Savage Attacks feature would also not work as it states:

When you score a critical hit with a melee weapon attack, you can roll one of the weapon’s damage dice...

The Barbarian’s Brutal Critical feature states:

You can roll one additional weapon damage die when determining the extra damage for a critical hit with a melee attack...

The Hunter Ranger’s Horde Breaker feature stats:

When you make a weapon attack, you can make another attack with the same weapon against a different creature…

And finally the Gloom Stalker Ranger’s Dread Ambusher feature stats:

If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of the weapon's damage type...

The above section of features only technically do not work with unarmed strikes as they are not specifically weapons or do not have weapon damage dice, however, making unarmed strikes count as weapons would allow all of these features to use unarmed strikes as well.

Allowing unarmed strikes to count as weapons would provide marginal benefits to a few features by allowing them to be activated even when no other weapons are available or when you would not want to use weapons. This a very small buff to a small subset of features and so it would not be imbalancing.