As far as the court case goes, it's basically just like it would be if the complainant had simply sued you directly for copyright infringement. If you win, the court can award you costs and attorney fees. I'm not sure how they decide whether or not to do so.
A claimant trying to use DMCA where it is not justifiable faces a couple other deterrents.
First, unless the claimant is representing themselves in the case, they are going to have an attorney, and that attorney is going to take into account Rule 11(b) of the Federal Rules of Civil Procedure [1]. If the claimant doesn't have a fairly reasonable case, he is going to have trouble finding an attorney.
Second, knowingly including false information on a DMCA takedown notice is perjury. The claimant can face criminal charges for that (although it would probably have to involve someone doing this on a large scale to get Federal prosecutors to prosecute).
Knowingly filing a false DMCA takedown also makes you liable for civil damages, including costs and attorney fees, incurred by the target of the takedown notice, the hosting provider, and in the case where neither the complainant or the target are the copyright owner, the actual copyright owner.
If the complainant who knowingly files a false takedown notice actually follows through and sue you for infringement, I'd guess that the damages due to you for the false notice would be handled there.
If the complainant isn't that stupid, and drops the matter after the notice and counter-notice, you could sue them over the false notice.
[1] https://www.law.cornell.edu/rules/frcp/rule_11