The rules for improvised weapons state that you can throw a melee weapon even if it does not have the thrown property for 1d4 damage:

If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage.

It is also stated that using an improvised weapon when it is similar to an existing weapon allows one to add their proficiency bonus to the attack:

At the GM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus,

Does this mean that when throwing a weapon this way, if a character is proficient with the weapon being thrown they get to add their proficiency bonus to this attack?

(The character does not have the Tavern Brawler feat which would make this trivial. This question asked in regards to a discussion on this homebrew question)

Only if you are ruled as using an improvised version of a proficient weapon

The relevant piece of information, PHB p. 147:

Improvised Weapons

Sometimes characters don’t have their weapons and have to attack with
whatever is close at hand. An improvised weapon includes any object
you can wield in one or two hands, such as broken glass, a table leg,
a frying pan, a wagon wheel, or a dead goblin.

In many cases, an improvised weapon is similar to an actual weapon and
can be treated as such. For example, a table leg is akin to a club. At
the DM’s option, a character proficient with a weapon can use a
similar object as if it were that weapon and use his or her
proficiency bonus.

An object that bears no resemblance to a weapon deals 1d4 damage (the
DM assigns a damage type appropriate to the object). If a character
uses a ranged weapon to make a melee attack, or throws a melee weapon
that does not have the thrown property, it also deals 1d4 damage. An
improvised thrown weapon has a normal range of 20 feet and a long
range of 60 feet.

Note the italicized line. It's not that every improvised weapon deals 1d4 damage when thrown, only those that "bears no resemblance to a weapon". And since your proficiency is based on the fact that you "can use a similar object as if it were that weapon and use his or her proficiency bonus", you cannot gain both benefits with the same weapon.

With this information, you effectively have two choices:

1. Have your DM treat the original weapon as a second, different weapon that has the Thrown feature. Example: treating a Halfling's War Hammer (no Thrown) as a Goliath's Light Hammer (with Thrown).




2. Make your attack as using an Improvised Weapon, dealing 1d4 damage and using the default thrown option of 20/60 range, and only adding proficiency if provided by another feature (as from Kensei or Tavern Brawler)

Using the base weapon as a thrown weapon:

Based on Jeremy Crawford's Sage Advice, weapons being used as an improvised weapon have none of their regular properties, unless the DM states otherwise. This ruling was made to answer whether a Ranged Weapon can still get benefits as a Ranged Weapon when being used for a melee attack (as when using Sharpshooter while bashing someone with a bow).

Using the weapon as an Improvised Weapon will treat it as if you're using a separate weapon and will no longer apply benefits specific to the original weapon, including your original proficiency bonus.

Yes

Assuming the DM rules that some weapon you are proficient in is “similar” to a weapon your “character is proficient with.” I have a hard time imagining how anyone could argue that this is not the case, since to do so would require claiming that the weapon is not an object “similar” to itself.

Relevant rules quotes:

If a character [...] throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

This shows that when you throw a non-thrown-melee weapon, it is an “improvised thrown weapon.”

In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club.

This is a possibility for any improvised weapon. A non-thrown-melee weapon that gets thrown counts as a improvised thrown weapon, which is to say it is an improvised weapon, and since it actually is a weapon, it is certainly “similar” to the weapon it actually is.

At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

And this shows that in the case of an improvised weapon being similar to an actual weapon, which almost-certainly must be the case here, proficiency bonus applies.

Irrelevant rules quotes

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object).

A weapon is not an object that bears no resemblance to a weapon. Therefore this sentence, in its entirety, does not apply, which is precisely why the next line, quoted above, says that throwing non-thrown-melee weapons “also” reduces their damage to 1d4.

If you use a weapon in a way that turns it into an improvised weapon—such as smacking someone with a bow—that weapon has none of its regular properties, unless the DM rules otherwise.

^{(Jeremy Crawford tweet)}

We aren’t discussing a regular property of the weapon—by definition we are discussing a property the weapon doesn’t regularly have, since we are talking about the thrown property of a weapon that doesn’t have it—so the fact that a weapon doesn’t have its regular properties is irrelevant. Jeremy Crawford does not deny an improvised thrown weapon its irregular thrown property, which the rules explicitly grant it.

Yes

The rules for weapon proficiency state (emphasis mine):

Proficiency with a weapon allows you to add your proficiency bonus to the Attack roll for any Attack you make with that weapon.

Since throwing a weapon you are proficient with is making an Attack roll with that weapon, you may add your proficiency bonus to the Attack roll.

To cite the rules cited in Daniel Zastoupil's answer:

In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

The bolded text makes it clear that using a ranged weapon for a melee attack or a melee weapon for a ranged attack does not cause that weapon to no longer be considered a weapon. When throwing a longsword at someone, you are still considered to be using a longsword. Using a weapon for an improvised purpose does not mean you should lose proficiency in it. Improvised weapons are weapons that were not considered weapons to begin with.

The damage reduction makes sense here as well - if you were to throw a sword designed for melee, it would stand to reason that the damage would not be as effective. As well, if hitting someone with a longbow, it stands to reason you would only deal 1d4 damage rather than the regular 1d8, since the 1d8 comes from the proper use of the weapon to fire an arrow.