Port 67: DHCP server
Port 67 is where DHCP servers listen, handing out IP addresses to devices when they join a network.
What runs on port 67
When a device connects, it broadcasts a request from UDP 68 to UDP 67, and the DHCP server replies with an IP address, gateway and DNS servers. BOOTP, the older protocol DHCP grew out of, used the same ports. Only one DHCP server should normally answer on a given network.
On a home network
Your router is almost certainly the DHCP server on your network and listens on 67. Pi-hole, some NAS boxes and mesh Wi-Fi systems can take over that role.
If port 67 is reachable from the internet
DHCP is a local-network protocol and is not useful across the internet, so a DHCP server answering on the WAN side would be a misconfiguration. On your own LAN, the thing to worry about is a second, unexpected DHCP server, called a rogue DHCP server, because it can send devices to a malicious gateway or DNS server.
How to check whether it is open on your network
Your router's DHCP settings page shows its address pool and current leases. If devices are getting addresses from an unexpected range, look for a second device acting as a DHCP server.
To find out whether port 67 is open on something in your home, and whether the internet can reach it, the app scans your Wi-Fi from your phone and checks your public address from the outside. It says what it checked and what it could not. Get RECON for iPhone — one-time purchase, no account, nothing leaves the device.