Sort of. First, we're not clear that the YouTube claim form is an actual DMCA claim (it's defective in a number of ways, not the least is the lack of a real counter-claim process).
Second, the 'perjury' part of the DMCA only applies if they falsely claim ownership of an infringed work. So if I make a bogus claim that your work infringes on something I actually have copyright over (however tenuous), it doesn't trigger the actual perjury bit. But if I claim to own, say, a Beatles album (which I do not) , or to be their copyright agent (which I am not), then it's a violation.
[EDIT: Just to clarify the above, this is to mean that I claim someone infringe on a Beatles album which I don't own or that I'm their agent when I am not. Thus, it's when my claim of ownership over the allegedly infringed upon work is a lie that this triggers, rather than any dispute over the ownership of the allegedly infringing work.]
I think that (unless barred somehow in YouTube's ToS), that a large enough group might be able to sue the groups making false claims for their ad money.
It's rather murky whether you could go after people who are, e.g., claiming copyright over public domain works, but with sufficient evidence of bad faith, maybe you could get somewhere.
The main problem is that federal lawsuits are really expensive and the amounts of money here are usually not all that large. It's not clear to me whether there are damages in an amount sufficient to make a class action worthwhile or anyone with deep enough pockets to pay a judgement even if one were obtained at great expense.
Or maybe someone could do a class action against YouTube as a whole for upholding bogus claims, but I'm not sure that you don't agree not to hold YouTube liable somewhere in some ToS and even if you do, I'm not sure if it's enforceable.
Disclaimer: IANAL, this isn't legal advice. If you have an actual legal issue, consult a lawyer, not random internet comments.