Fans of Café Gray’s spiced venison loin and skate schnitzel may want to make a reservation promptly at that Time Warner Center restaurant. Amid a shaky economic climate, Gray Kunz’s upscale café, popular with Manhattan’s social set, will close its doors for good at the end of June after a three-and-a-half-year run, according to restaurant staffers. When the eatery opened in October 2004, it earned mostly positive reviews for the food — and mostly negative ones for its open kitchen, which gave cooks glorious views of Columbus Circle and Central Park, but largely denied them to its patrons. A spokeswoman for the restaurant did not return calls, but receptionists, who did not give their names, at Café Gray and at Mr. Kunz’s 7-month-old Midtown eatery, lounge, and event space, Grayz, confirmed the Time Warner Center eatery’s June 30 closing date — and that there are no plans to reopen.