What is a word for fake seriousness, especially when you are intending to be funny

Solution 1:

The word deadpan is probably the best fit. It can be used as an adjective or adverb (or a noun or a verb):

adjective

marked by or accomplished with a careful pretense of seriousness or calm detachment; impassive or expressionless:
deadpan humor.

adverb

in a deadpan manner:
He spoke his lines utterly deadpan.

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Based on the Random House Dictionary, © Random House, Inc. 2018.}

Solution 2:

straight faced TFD straight′-faced′ (-fāst′) adj. straight′-faced′ly (-fāst′lē, -fā′sĭd-lē) adv.

A face that betrays no sign of emotion.

As in:

He was a very straight-faced prof, but he has a tremendous sense of humor.

She was watching me totally straight-faced and it took a couple of seconds before it dawned on me that I'd just been had.

And here is an article on some famous straight-faced comedians: article

Solution 3:

The word that came to my mind was "dry wit". A search of "dry wit" will bring you to Laurel's answer. Laurel's answer is better anyway, and more common.

dry
14. Humorous in an understated or unemotional way: dry wit.
definition of dry