Are there some techniques which can be used to show that a sum "does not have a closed form"? I am aware that there are some techniques which can be used to show that some function does not have an antiderivative expressible using elementary functions, such as Liouville's theorem. (More broadly, this falls into the area of differential Galois theory and differential algebra. Such topics were discussed also on this site, for example, here or here.)

Are there some analogous results for finite sums? More precisely, suppose we have a sum $$s(n)=\sum_{k=1}^n f(k)$$ where $f$ is a given elementary function. Are there some methods which can be used to show that $s(n)$ is not equal to an elementary function (restricted to $\mathbb N$)?

To give a specific example, let us take $f(n)=\frac1n$. Are there some methods which can be used to show that we cannot express the $n$-the harmonic number $H_n=\sum_{k=1}^n\frac1k$ as $H_n=s(n)$ for an elementary function $s(x)$? (And is such result for harmonic numbers even known?)
EDIT: Very shortly after posting this question I found this: Do harmonic numbers have a “closed-form” expression?
Which definitely answers the second part about harmonic numbers. (Gerry Myerson's answer mentions also others sums, not only $H_n$. So it can definitely be considered as an answer for this question, too. We will see whether somebody will post some additional interesting information as an answer to this question.)

Look for "Hypergeometric Summation", "Summation in finite terms" and "Symbolic summation".
There are parallels to the Risch algorithm.
Read e.g. the chapter "Symbolic Summation" in Bona, Miklos: Handbook of Enumerative Combinatorics. Chapman and Hall/CRC 2015.
There is a theory or an algorithm of Michael Karr:
Karr, Michael: Summation in finite terms. J. Assoc. Comp. Mach. 28 (1981) (2)305-350
Karr, Michael: Theory of Summation in Finite Terms. J. Symbolic Computation 1 (1985) (3) 303-315
Karr's algorithm is the analogue of Risch algorithm for summation.
And there is a theory or an algorithm of Carsten Schneider:
Look for
Schneider Summation
and for
Schneider sums
