Whilst I agree with the answer provided by user8719 in that the seven kings are not specific and given Tolkien's reliance on the number 7 for several things, from seven kings of the Valar and seven queens of the valar to seven dwarf-lords and seven seeing stones for example (which I appreciate has also already been pointed out), it could be any group of kings being referenced by Saruman.
However, in the context of Saruman's dressing down of Gandalf's "power grab," as he sees it, I find it rather unusual that Saruman wouldn't have a specific reason to mention "crowns of seven kings." He mocks Gandalf asking if he would bid Saruman to hand over specific items of great value or importance. Most of these are fairly straight forward, the key of Orthanc (Saruman's tower and seat of the Istari given Saruman's position as leader of the five wizards), the keys of Barad Dur (Sauron's ancient fortress where he currently dwells), and the rods of the five wizards (namely Saruman, Gandalf, Radagast, Alatar and Pallando). These all make reference to specific items that are in play within The Lord of the Rings story directly.
Both Orthanc and Barad Dur are the primary abodes of the villains of the story (Saruman and Sauron), the five wizards are directly (or supposed to be) in direct opposition to Sauron and working to rally the free peoples against him (only Gandalf succeeds in this task as Saruman falls, Radagast focuses on nature (though his purpose is less clear due to the Valar that sent him), Alatar and Pallando journeyed east into the lands of Rhun (home of the Easterlings) and it is unknown what became of them - though given that the Easterlings rallied to Sauron's call, the blue wizards were likely either corrupted or killed). Therefore these items are in direct reference to elements that are ongoing with the story of the Lord of the Rings rather than referencing ancient persons or artifacts.
When the "crowns of seven kings" are mentioned it is very easy to immediately be drawn to the ring verse and the mention of the seven rings given to the dwarves. I admit upon my first reading of it, that's where my first thoughts went, but, having thought about it the ring verse actually is pretty clear in its terminology. "Three rings for the Elven-Kings," "Seven for the Dwarf-Lords," and "Nine for Mortal Men." So immediately there in the text Tolkien specifically mentions the Dwarves as Lords not Kings. Now, I know it's basically nitpicking, but in Tolkien's case he was a linguistics professor, his use of words were just as important to him as anything else. In that respect given that Tolkien makes specific mention of Elven-Kings, why not say Dwarf-Kings if that was the intended implication? Instead he says "Dwarf-Lords." Thus it seems to entirely discount the Dwarf-Lords as having crowns of any kind, let alone be relevant enough for Saruman to consider them important to mention in mockery of Gandalf becoming head of the Istari.
So then, as you mentioned, could it be a reference to Gondor because of Gandalf's song that Pippin hears? But, as has been pointed out, at best you could draw four Gondorian or rather Numenorean Kings by including Arnor and its broken down components, but then that's three short, even if you were to include other free peoples of the race of men such as Theoden of Rohan it still falls short. So Gondor doesn't really fit unless it were the crowns of seven previous kings of Gondor, which might be relevant for Gandalf intending to crown Aragorn King, but then why would Gandalf need seven crowns to coronate Aragorn? Surely one would be enough. Maybe if one were Earnur's crown or Arvedui's, but then why would Saruman not be more specific? So again it doesn't seem to be relevant given the context.
Looking at some of the other possibilities like the potential crowns of the Valar for example, it seems a little far fetched for Gandalf to go to Saruman to obtain the crowns of beings vastly more powerful than Saruman and more than likely still in the possession of their owners. Whilst Saruman is referencing items he doesn't technically have, namely the "rods of the five wizards," they are still under his power and direct influence until Gandalf casts him out of the order of Istari. Therefore, it would seem strange for Saruman to mock Gandalf by offering to fetch him something that is beyond Saruman's power to offer. This goes the same for the Palantiri. Whilst Saruman did posses one, we know for a fact Denethor had one in Minas Tirith which allowed Sauron to drive the steward to madness and despair. Sauron himself had claimed one, which he used to corrupt Saruman and torment Denethor, which had been gifted to him by the Nazgul after the fall of Minas Ithil. The others were either lost or marred. Gandalf himself mentions that the stones were lost. In addition, whilst the palantiri are often referred to as the seven seeing stones, there was at least one other, the Master Stone, in the Tower of Avallone in Tol Eressea and so the seven only refers to the palantiri present in middle earth.
As for the dwarf fathers, there is a more compelling case here given they each founded a specific line of dwarves. Though again, it seems peculiar as to why Saruman would possess a crown for any of these dwarves or why he would see it as a specific thing to mention to Gandalf. Where the other three have a relevance to the story and could be of use to Gandalf, hence Saruman's choice to mock Gandalf with them, the crowns of the dwarven fathers don't really aid Gandalf. The dwarves are already committed to the war against Sauron following their attendance at the council of Elrond. Furthermore, the dwarf fathers were sacred to the dwarven peoples, so much so, their tombs were jealously protected as the dwarves believed the spirits of the dwarven fathers would return to their bodies and so the dwarves purposely preserved them. In this instance, for Saruman to gain even just one heirloom from any of the dwarven fathers seems to be highly unlikely.
The sons of Feanor is another interesting possibility, but again, their relevance to Saruman and Gandalf in this moment is lacking. Not only this but not all of Feanor's sons ruled as kings of the Noldor and their stories were far more entwined with the silmarils than their succession to Feanor's crown. Maglor, the only supposedly surviving son of feanor, was not crowned as a king, nor were his brothers Amrod and Amras who didn't live long enough to be crowned as kings due to their deaths in the third kinslaying. Whilst there were all princes, they were not all kings.
So considering then that this dialogue is a mockery of Gandalf by Saruman with items that could be considered useful to him. The key of Orthanc gives Gandalf the Tower of the White Wizard, the keys of Barad Dur allow Gandalf direct access to Sauron, the five rods of the wizards give Gandalf dominance over the Istari, the seven crowns must therefore have some usefulness to Gandalf. This could be a reference to seven Gondorian Kings and the line of succession to crown Aragorn as King, something Saruman despises and would mock Gandalf over, though why would he offer Gandalf seven when one would suffice? There are also no specific Gondorian or Arnorian kings mentioned elsewhere that these crowns could belong to.
The only other possibility that could be considered is if it is in some way connected to the Nazgul, the third antagonist faction beside Sauron and Saruman. With the other three offered items, Saruman is referencing both himself and Sauron so in some ways it would make sense to reference the Nazgul who have been plaguing Frodo since he left the Shire. Whilst there are nine Nazgul, it is not specifically mentioned by Tolkien that all these men were once kings. The ring verse itself only refers to "Nine for Mortal Men doomed to die." Tolkien was also vague with the identities of the Nazgul, naming only two, the Witch-King of Angmar and Khamul the Easterling. The rest are described as "mighty in their day, kings, sorcerers and warriors of old" and we are also made aware that three of their number were Numenorean - with the Witch-King being implied to be one of these Numenorean's.
Given Tolkien's vagueness around the identities of the Nazgul, even his named Nazgul, it is possible that only seven of them were ever kings in their time. All were powerful men, as Sauron sought out power that he could corrupt, but Sauron was also devious. It would make sense for Sauron to seek to corrupt those that could attain power and influence that he himself could then dominate. He also didn't focus solely on one group of people. Khamul, the only other named Nazgul, is specifically mentioned as being an Easterling, thus it would make sense for Sauron to have targeted powerful leaders in the realms of Rhun, Harad and Khand where he had considerable support during the Third Age.
Now, whilst this is certainly an interesting possibility and Gandalf coming into possession of seven crowns that could have once belonged to Nazgul is intriguing, it is by no means certain. Whilst what we know of the Nazgul does allude to a few of them not being Kings, we have no specific number. Perhaps only five of them were kings, or maybe even only two, the fact is Tolkien didn't say. If the seven crowns are connected to the Nazgul it certainly adds more to their mystery and makes some sense in Saruman's mockery of Gandalf. How could Gandalf utilise the crowns of seven Nazgul? Would naming them grant Gandalf an influence with them? Who knows. But it is clear that Saruman intends to sneer at Gandalf assuming his position and the things he decides to mock Gandalf with pertain to things that Saruman seems to believe Gandalf would find a value of power in.
I agree the "Crowns of Seven Kings" are from seven unspecified kings, but could it be possible that they have a deeper connection to the opponents of the free peoples? It certainly swings that way under the context of the dialogue. Whether it relates to the Nazgul or not, we probably won't know unless we somehow unearth any more commentary from Tolkien himself.