Here's a standard theistic argument against Platonism: If Platonism is true, then God is dependent on properties like divinity, goodness, omniscience and omnipotence. But God is not dependent on anything. So, Platonism is false.
I think it's worth noting that this argument only works given heavy-weight Platonism. The light- and heavy-weight Platonists agree that, at least if F is fundamental, x is F if and only if x instantiates Fness. But the heavy-weight Platonist adds the claim that if x is F, it is F because it instantiates Fness. The light-weight Platonist--van Inwagen is the most prominent example--makes no such explanatory claim.
Without the explanatory claim, the dependence argument for a conflict between Platonism and theism fails. For while it may be true on light-weight Platonism (assuming "is divine" is fundamental--something that Jon Jacobs at least will deny--or an abundant Platonism) that God is divine if and only if God instantiates divinity, we cannot conclude that God's being divine depends on God's instantiating divinity or on any other property. Indeed, the light-weight Platonist could (but does not have to) even make the opposite claim, that God instantiates divinity (or goodness, omniscience and omnipotence) because he is divine (and good, omniscient and omnipotent).
Of course, the aseity argument isn't the only reason to deny Platonism. God is the creator of everything other than himself, and that causes problems for properties, too.