No, a dragon's breath cannot, in general, pass through Leomund's Tiny Hut
The key line of the spell that prevents it is emphasised in this quote:
Creatures and objects within the dome when you cast this spell can move through it freely. All other creatures and objects are barred from passing through it. Spells and other magical effects can't extend through the dome or be cast through it. The atmosphere inside the space is comfortable and dry, regardless of the weather outside.
It is important to point out that the modifier added to the end of the last sentence in that quote:
[...] regardless of the weather outside.
Means the last sentence applies to "weather" from outside the hut affecting the "atmosphere" inside the hut. Notably, it means the sentence does not cover things originating inside the hut from changing the "atmosphere" inside the hut. All effects should be viewed with this lens applied. For our purposes, the dragon's breath weapons definitively originate outside of the hut, and are thus caught by this sentence.
Restrictions on dragon's breath weapons entering the hut as a result of this last sentence
First things first, "weather" is defined as:
the state of the atmosphere with respect to heat or cold, wetness or dryness, calm or storm, clearness or cloudiness
disagreeable atmospheric conditions: such as
- RAIN, STORM
- cold air with dampness
Next "atmosphere" which is defined as:
the air of a locality
the gaseous envelope of a celestial body (such as a planet)
So, in particular the gases inside the hut will always be comfortable and dry. The types of dragons breath we have currently in the game can be summarised as follows:
- Some form of gas (Green, Brass, Faerie, etc)
- Acid (Black)
- Fire (Red, Gold)
- Lightning (Blue, Bronze)
- Icy Blast (White, Silver)
The gas forms of breath are immediately excluded, as they would make the atmosphere in the hut decidedly uncomfortable (unless the caster is immune to the specific gas).
Acid is also excluded as, by definition, acid is:
a sour substance
specifically : any of various typically water-soluble and sour compounds that in solution are capable of reacting with a base (see BASE entry 1 sense 6a) to form a salt, redden litmus, and have a pH less than 7, that are hydrogen-containing molecules or ions able to give up a proton to a base, or that are substances able to accept an unshared pair of electrons from a base
This is excluded explicitly by virtue of the requirement to that the interior of the be dry regardless of what is happening outside. So the hut is clearly capable of blocking liquids, and getting doused by a liquid is the very definition of being wet:
consisting of, containing, covered with, or soaked with liquid (such as water)
The Icy Blast is also excluded, as it causes cold damage by drastically lowering the temperature. We can see this from the description of the White Dragon:
[...] Larger treasures and chests are encased in layers of rime created by the white dragon's breath, and held safe between layers of transparent ice.
I don't know about you, but temperatures cold enough to cause accumulations of a crust of ice on objects would be a decidedly uncomfortable environment to be in (unless you are immune to the cold). They are also firmly in the definition of "weather" presented above.
That leaves us with Fire and Lightning breath. We know the dragon's breath weapon is not magical (as confirmed by an official Sage Advice ruling), as a result the Fire and Lightning produced respectively is not magical Fire or Lightning. This means it needs to conform to the regular rules for such occurrences.
So we need to ask:
What is fire and what is lightning?
Fire is defined as:
fuel in a state of combustion (as on a hearth)
For fire to exist, three things are required, Oxygen, fuel and heat. We clearly have heat and Oxygen, so that leaves the fuel. Thus, in order to exist as fire in the hut the fuel the dragon's breath is burning must travel with it into the hut. The fuel itself is certainly an object (or at least each element of fuel is an object) and as a result cannot pass into the hut. All that remains is the heat. Again, I don't know about you, but being in a hut with air hot enough to cause significant damage is not my idea of a comfortable atmosphere! Additionally, "heat" falls under the definition of "weather" outside. Thus the fire breath is blocked by the hut.
Lightning is defined as:
the flashing of light produced by a discharge of atmospheric electricity
Here the definition contains the key word, atmosphere! An electrified atmosphere is not a comfortable one for most forms of life! Thus we must conclude that the hut also blocks lightning breath.