It seems that Zabbix Server does not clean-up database sessions after establishing a successful connection where the DB is not usable for some reason (permission issues, schema issues, etc.).
The number of sessions will increase indefinitely until the maximum number of sessions is reached on the DB server.

Steps to reproduce:

On a working environment, revoke the 'public' permissions to schema 'public' on the zabbix database:

Result:
1. Check the server log for this error:
21381:20180204:160045.582 [Z3005] query failed: [0] PGRES_FATAL_ERROR:ERROR: relation "users" does not exist
LINE 1: select userid from users limit 1
^sessions
[select userid from users limit 1]
21381:20180204:160045.582 cannot use database "zabbix": database is not a Zabbix database

2. The number of postgress TCP sessions (on the Zabbix Server's side) and postgres sessions (on the DB server's side) will increase slowly until the PostgreSQL max_connections limit is hit.

3. There is no visible change in the number (nor PIDs) of Zabbix Server processes.