Ansvaret føles som en byrde.

Because -et is the definite ending for many neuter nouns in Norwegian.

• et ansvar = a responsibility
• ansvaret = the responsibility

So instead of putting a separate word like the before the noun, Norwegian usually adds the definite article to the end of the noun.

Here, føles means feels in the sense of seems / is experienced as.

This is a very common Norwegian pattern:

• Det føles bra = it feels good
• Det føles rart = it feels strange
• Ansvaret føles som en byrde = the responsibility feels like a burden

By contrast, føler is often used when someone actively feels something:

• Jeg føler meg trøtt = I feel tired
• Jeg føler sorg = I feel sorrow

So in this sentence, føles is the natural choice because the subject is the thing being described.

It looks like one, because -s forms in Norwegian can be passive, but here it is better understood as an s-verb or middle-style form.

So although it resembles a passive, in this sentence it simply means something like:

• feels
• seems
• comes across as

In other words, Ansvaret føles som en byrde is not really about someone actively doing the feeling to it. It describes how the responsibility is experienced.

Here, som means like.

It introduces the comparison:

• føles som en byrde = feels like a burden

This pattern is very common in Norwegian:

• Det ser ut som regn = it looks like rain
• Det høres ut som en god idé = it sounds like a good idea
• Det føles som en drøm = it feels like a dream

Because byrde is a countable noun, and after som, Norwegian usually uses the indefinite article when comparing something to one example of that thing.

So:

• som en byrde = like a burden

That is the natural phrasing here.
Just som byrde would usually sound unnatural in this sentence.

Also, byrde is a common gender noun:

• en byrde = a burden
• byrden = the burden

Because in Norwegian, the definite article is usually attached to the noun itself.

So:

• et ansvar = a responsibility
• ansvaret = the responsibility

English uses a separate word, but Norwegian often uses an ending instead.

They are:

• et ansvar = a responsibility
• en byrde = a burden

Their common forms are:

• et ansvar / ansvaret
• en byrde / byrden

You may also see plurals such as:

• ansvar / ansvarene
• byrder / byrdene

But in this sentence, both nouns are singular.

Not especially. The sentence follows a very straightforward pattern:

• Ansvaret = subject
• føles = finite verb
• som en byrde = comparison phrase

So the structure is:

Subject + verb + complement

This also fits the normal Norwegian V2 rule, where the finite verb comes in second position in a main clause.

Yes, but it means something slightly different.

• Ansvaret føles som en byrde = the responsibility feels like a burden
• Et ansvar føles som en byrde = a responsibility feels like a burden

The first refers to a specific responsibility.
The second is more general or less specific.

So if you are talking about one particular responsibility already known from context, ansvaret is the right form.

Yes, it is natural and idiomatic.

It sounds like normal written or spoken Norwegian when describing something emotional or psychological. It is a good example of how Norwegian uses føles som to talk about subjective experience.

Similar natural sentences would be:

• Jobben føles meningsløs
• Situasjonen føles tung
• Presset føles som for mye