Sorry for reviving this old thread, but because it is (currently) the first result, when searching for “proxmox home assistant static ip”, I wanted to add something to consider… 
@Tromperie is basically right, that assigning a static lease is the most elegant and from a network administrator’s perspective most “correct” way assigning IP’s in your local network. In fact, I’ve done this for years, ignoring the issues associated with this approach, just because static IP’s are, you know, … “evil” 
But finally I came to the conclusion, that, in real life, you spare yourself much trouble, assigning static IP’s for the most essential two or three devices in your local network on the devices itself and also using IP’s instead of FQDN’s for the core configuration of these servers, i.e. using an IP instead of a hostname for the HA recorder connection string confguration.
Just pay attention to not assign these static IP’s to more than one host. F.e. keep track of them as commented out entries in your dhcp and /etc/hosts config.
The big advantage of using static IPs for such devices is, that they are not affected by dhcp or dns server problems on your network, i.e. a dns and/or dhcp server going dark or running wild for some reason.
In the past few years I had more than enough trouble with unstable pihole instances, stubborn routers and the like. Especially HAOS has a bad habit in constantly trying to mess up local name resolution, as some of you might know.
Since using a static IP for my HA instance at least my home automations are still running fine, even while debugging the next obscure dhcp or name resolution problem appearing out of thin air.
I can assure you, a robust HA server is greatly improving the WAF for your home automation… 