Is this question without the verb to do valid or does it have to be "Where does the problem lie."? These questions should require the to do but this particular one doesnt sound horribly wrong so I wanted to check on its correctness as I hear them sporadically. If it is grammatical I would also like to know which other verbs dont require the auxiliary to do.
In older English—roughly, through the transition from Early Modern English to Modern English, 1500-1700—any verb might invert with its subject to mark a question, so do support was not required to supply an invertible auxiliary.
The use has never completely died: it was frequent in literary contexts well into the 20th century, and it lingered with verbs like go, run, lie denoting states into the 19th—Nelson famously asked Hardy "How goes the battle? How goes the day with us?" as he lay dying at Trafalgar. You'll still encounter the greeting "How goes it?" from time to time.
But today the use is markedly literary and old-fashioned. I advise you to avoid it in ordinary discourse.