I have actually seen government agencies complain about being deluged by FOI requests, the cost of dealing with them etc. They mostly get ignored because on inspection the "deluge" of FOI requests tends to be from journalists digging for stories, and that's sort of what we want them to do. Also because the high cost of FOI responses tends to reflect messy and disorganised internal information systems rather than anything fundamental.
That said, I don't think it's really comparable to the GDPR. For one FOI compliance is a joke, organisations get out of it all the time on the thinnest of pretexts. There's no real incentive for a government to police itself in this regard. But GDPR enforcement is incentivised by large sums of money, for an organisation that is technically bankrupt.
