I cannot help but think about the verb なる when I hear or read the expression ~てならない.
Are the verb and the expression even related?
The expressions are related: the ならない in ～て・ならない is simply the negative form of なる.
However, it's worth noting that なる means more than just "to become": the underlying meaning is more like "to come into being". If I've understood correctly, current linguistic studies by Bjarke Frellesvig suggests that なる is related to an ancient copula (like English "is, to be") starting with n-.
Putting this together, another way of looking at this expression is
[VERB]て・ならない → "
[VERB] and it won't be" → or more idiomatically as English: "(if / when)
[VERB] happens, it won't do". The "it" here is the generic "it", like the "it" in "it's raining", or when you ask someone "how's it going".
何【なに】かして（は）ならない works out to "doing something won't do", and is an expression of prohibition: "you mustn't or shouldn't do something".
淋【さび】しくて（は）ならない works out to "(so) sad that it won't do" → "too sad (to go on, to continue with something, etc.)".
If the part before the ならない is itself in the negative, as in 何【なに】かしなくて（は）ならない, it works out to "not doing something won't do" → "you must or should do something".
What is the origin of the expression ~てならない ?
In a nutshell, this is ancient, and possibly older than Japanese itself.
I've read that this kind of construction for "must" or "should", using a negative conjunction and a negative form of the verb for "to become / to come into being", is shared by other languages in the region. That said, I don't speak or read Korean, Mongolian, Manchu, Tungus, ... so I cannot evaluate these claims myself.
If this construction is common to other languages in the region, that might corroborate the Altaic theory that Japanese is related to specific other languages, and it would suggest that this construction was inherited into Japanese from some earlier ancestor language.
Regardless of language relations, the て・ならない expression just within Japanese is very old indeed, appearing in the oldest written records in the Japanese language, such as the Man'yōshū completed some time around 759: see examples here, using the Old Japanese negative form of the verb, ならず.