I'm having the same issue. My dhcp lease time is set for 24H, but the apple tv asks for a new lease about once a minute.
Logs:
Jul 7 11:40:41 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:41:54 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:41:54 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:43:06 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:43:06 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:44:18 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:44:18 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:45:30 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:45:30 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:46:16 unknown daemon.info dnsmasq-dhcp[531]: DHCPINFORM(br0) 192.168.1.208 fc:aa:14:72:37:b0
Jul 7 11:46:16 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.208 fc:aa:14:72:37:b0 blackbox
Jul 7 11:46:42 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:46:42 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:47:54 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:47:54 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:49:05 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:49:05 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:50:41 unknown daemon.info dnsmasq-dhcp[531]: DHCPREQUEST(br0) 192.168.1.206 1c:1a:c0:79:29:9e
Jul 7 11:50:41 unknown daemon.info dnsmasq-dhcp[531]: DHCPACK(br0) 192.168.1.206 1c:1a:c0:79:29:9e Apple-TV
Jul 7 11:51:20 unknown daemon.info dnsmasq-dhcp[531]: DHCPINFORM(br0) 192.168.1.200 4c:0b:be:0b:a5:a0