Can, for instance, a character with no special unarmed combat abilities attack an opponent wearing full plate and wielding a long sword with a bare-fisted punch without any drawbacks, risks or consequences?
There are no drawbacks, risks, or consequences specifically for using an unarmed strike against any enemy, regardless of their weapons or armour.
The unarmed strike is a melee weapon attack that all characters are proficient with, so you will never have the drawbacks that come with using a weapon you're not proficient with.
Finally, while the damage of unarmed strike is just 1, you still add your Strength modifier to its damage, so if you've maxed Str you're still doing 6 damage on every attack. Obviously other weapons are generally better, but if you're caught out without a weapon the unarmed strike is definitely worth doing.
There is nothing in the rules that prohibits or punishes an unarmed attack.
However short of being a monk or having the Tavern Brawler feat, it does a laughable 1 damage unless you have unarmed proficiency (which is then 1 + STR mod).
Druids and Wizards do not have proficiency as they are not proficient with simple melee weapons (wild-shaped druids being an exception).
Your question comes from not understanding the concept of "health" in DnD. Loosing Hit points is not exactly getting wounded. Hit points is a measure of you fighting alertness, health and fatigue altogether. If A hits B and takes some hit points from B, this could mean that A deals a scratch, or simply makes flourish that makes B to exert himself when dodging it. This is not governed by numbers or texts at all, only by the play situation.
So if brawling with armored enemy, you do not hit him every turn into steel, chipping away little parts of his body. You jump, dodge, push, pull, tease, insult and wear down your enemy in all possible way. Only when you take last HP, you deal a blow that connects. It can be something dramatic like "suddenly you see a fork on the floor, grab it and thrust deep into enemy's helmet. With a loud bang, he drops down". Same with you: you are fatigued, got bruises, but did not get three flat hits with a mace in the head.
This understanding of mechanic explains a lot: scars, being able to fight as good as new till last HP, the "bloodied" state from DnD4, strange fighting situation, like beating stone golem with a whip, and many others.