iptables - How to allow all connections to the local lan?

As llua said, using ipset. However, you'll probably want to make this persist through reboots; ipsets are stored in memory otherwise.

ipset save > /etc/ipset.conf, then enable the ipset service so it recreates the set on reboots (on Arch systems, at least, which provide an ipset.service systemd file).

Add to /etc/rc.local the line ipset restore -f /etc/ipset.conf on most other Linux systems.

See the always-excellent Arch wiki page for more.

Using ipset

ipset create locallan hash:net
ipset add locallan 192.168.0.0/16
ipset add locallan 172.16.0.0/16
ipset add locallan 10.0.0.0/8

iptables -I INPUT -m set --match-set locallan src -j ACCEPT

Would allow connections from those ranges to the server with that rule.