Locke thinks there are no such things. He notes that the argument usually given in support of innate ideas is that there is universal agreement by men on some principles, and that this proves them to be innate, known naturally and automatically.
His first point is to note that universal agreement does not prove that they are innate principles; for it is still possible to be known by some other way by everyone.
His second point is that there are not any principles which are universally assented to. "Whatever is, is" and "it is impossible for the same thing to be, and not to be" are supposed to be examples of innate ideas, but these are not universally assented to by all creatures with minds: idiots, children, and savages don't agree with them, and we find no evidence of such beliefs in them.
He thinks it is essential to the notion of an innate idea that it be known by the person to whom it is innate; an idea imprinted upon a mind which is ignorant of it makes no sense for Locke; it is tantamount to knowing a thing and not knowing it at the same time. And if it be possible that a principle or a proposition be imprinted upon a mind which all its life remains ignorant of it, then why not think every true proposition is innate? (This problem of distinguishing which propositions are innate and which are not is one that Locke brings up often.) So an innate idea must be one that a person be aware of, and know. But there are plenty of human minds which have no awareness of principles fancied by some to be innate, like the two named above; so because there is no universal assent on these principles, they are not innate to human minds.
The usual response to this, Locke says, is that such principles are universally assented to when men come to the use of reason. But Locke is not enchanted by this response to his argument. There are two ways this response is to be understood: either it is that once a person has the use of reason, he comes to be aware of these principles and truths and assents to them, or else that once a person has the use of reason, they are discovered by him.
If the second, the fact that they are discovered by reason shows them not to be innate. By this way of thinking, all the truths of mathematics would be innate--which they are clearly not! Further, the fact that they are discovered by reason (ex hypothesi) proves them not to be innate. Reasoning, says Locke, is just the use of that faculty of the mind by which we discover previously unknown truths by way of known truths; but if we discover an idea by reason, it follows that it cannot be innate. Further, everything we ever know by way of reason (a good number of truths, as you can imagine) would turn out to be innate, which is an unsavory consequence.
The possible response is that the deductions of the mathematicians are not self-evident and assented to in the manner that the maxims they use are, and this is a mark of the innate ideas. But the maxims are not by way of reasoning; we don't reason to them, but reason by them, and so that innate ideas are discovered by reason turns out to be false.
If it turns out that the proponent of the doctrine of innate ideas instead meant that these principles are discovered by us at the time when we are able to use reason, the first of the possible interpretations, then this is simply false and frivolous, by Locke's account. False, because there are plenty of children who obviously have the use of reason and do not assent to the two examples given above of innate ideas. Further, how do we distinguish between those truths which are known when we come to the use of reason which are innate and which are not? Frivolous, because how on earth is it supposed to follow from the fact that our faculty of reason is capable of use that we thereby know these truths, any more than from the fact that our faculties of speech or sight are capable of use?
The suggestion that universal assent when the respective principles or propositions are understood proves them to be innate proves too much; for too great a number of propositions are such that, once the terms are understood, anyone would agree and assent to them. Any proposition of the form "A is not B", where A and B are distinct concepts of ours, would have to be innate because it would be assented to by anyone who understands the terms; thus the proponent of the doctrine of innate ideas would have to allow as many innate ideas as we have human concepts and possible formulations of sentences of that structure. And that's just too many.
Nor let it be said, says Locke, that such particular instances (like "yellow is not bitterness") are assented to because they are entailed by more universal and general principles which are themselves innate ideas; for it is clear that the thought that A is not B, where A and B are given particular objects, occurs in our minds earlier than any such general proposition. Children wouldn't assent to "A is not B, where A and B are distinct concepts", but would agree with "sugar is not lemon".
Locke then proposes another argument. Suppose we have an innate idea, a proposition. What parts of it are supposed to be innate? For persons only know what words are and mean when they are told by others, see the referents pointed to, etc. The referents of the nouns, then, require knowledge by experience, which is clearly not innate. But not only are the names in the sentences not known innately, but neither are people born with ideas of lemons, numbers, cats, red, yellow, etc. These things are learned as well. But then what is left to be innate? It seems nothing at all!
So Locke concludes his argument: if there were innate ideas, they'd be universally assented to by all human beings with minds. But since there is no universal assent to such ideas, and because there are no helpful ways of distinguishing which ideas would be innate and which would not, there are no innate ideas.
He gives another. He says that we'd expect innate ideas to be most clearly understood and grasped by those with very little contact with other persons. He is worth quoting at length here, because his writing is funny:
For children, idiots, savages, and illiterate people, being of all others the least corrupted by custom or borrowed opinions; learning and education not having cast their native thoughts into new moulds, nor by superinducing foreign and studied doctrines confounded those fair characters nature had written there; one might reasonably imagine that in their minds these innate notions should lie open fairly to everyone's view, as it is certain the thoughts of children do... One would think, according to these men's principles, that all these native beams of light (were there any such) should in those who have no reserves, no arts of concealment, shine out in their full lustre, and leave us in no more doubt of their being there than we are of their love of pleasure and abhorrence of pain. But alas! amongst children, idiots, savages, and the grossly illiterate, what general maxims are to be found? what universal principles of knowledge? (I.II.27)
Thus John Locke goes all out against the doctrine of innate ideas. He continues his discussion with the belief that there are some innate practical principles, a section to be summarized in a later post.
http://snemes.blogspot.com/2011/01/locke-against-innate-ideas-part-i.html