My parents have been socially separated for years but legally married. Is my mother responsible for his taxes?

aside from the fact that his final arrangements caused you expense , in a way , he didnt need a penny to his name after he expired .
i would like to leave my sons " things " but in reality , i was building my first house with a chainsaw at the age of 27 . if they dont have that kind of drive , maybe they deserve to rent as long as they live .
the only thing i ever got from my dad was scorn -- because i didnt believe in the same sky - god that he did .
ive had my reward here on earth . dad might get his later -- or he might trade it all for a sump pump and some worm repellant ..

Oh and there is no executor of the estate because there is no estate. Sorry, forgot to include that in my last post.

The only thing he had that was taxable was the retirement account. That is where the $150 owed is from. However, she had no taxable income and I filed his taxes separate from her, not jointly.
He had no remaining assets. No money in the bank, no death benefits, nothing. I have personally paid for everything after his death such as the funeral, etc. He had a credit card in his name that was written off because there is literally nothing to get from him (no estate).
The remaining money in the retirement account went away because no one was named as a beneficiary.
Sad situation to see someone pass away and have not a penny to his name.

Well he had something taxable or there would not be a bill. The Executor of the estate has to make sure it is paid from the remaining assets. Like the money he had in the bank, perhaps. Or the death benefit from a pension plan.
I'm going to assume the remaining retirement account went to mom outside of any probate process, as a beneficiary. That will generate income. So she has an asset IRS can go after.

the irs makes a lot of flap and junk mail . its the state who will slap liens on everybody who ever associated with you .

She had no refund. The only income she has is social security and did not receive enough to have to file taxes.

Yup, the IRS will take it out of her refund, because they are still "married". Hate to be the bearer of bad news.