Can taking buspirone cause a gout flare?

Not usually. Buspirone should not cause a elevation in your uric acid. Buspirone has a side effect of kidney imparment. If you have this side effect it might effect your uric acid and gout.
Probably not. There is nothing in the medical literature that Buspirone directly causes a gout flare. Neither epocrates nor uptodate list gout as an adverse effect. There could be an indirect relationship in that Buspirone could adversely affect kidney function and this could indirectly reduce clearance of uric acid, thus allowing it to build up.