Bind process or user to specific IP (Linux)

iptables -t nat -A POSTROUTING -m owner --uid-owner user1 -j SNAT --to-source 192.168.1.2
iptables -t nat -A POSTROUTING -m owner --uid-owner user2 -j SNAT --to-source 192.168.1.3
iptables -t nat -A POSTROUTING -m owner --uid-owner user3 -j SNAT --to-source 192.168.1.4

It is your responsibility to make sure that (a) you are not otherwise using the POSTROUTING nat chain, so these rules don't conflict with anything else, and (b) all these IP addresses are present on your NIC (you won't hear many replies otherwise).

This will also only affect traffic originating locally from processes owned by these users. If these are users are setting up network listening daemons, a different approach will be needed to handle replies, and if the server is acting as a router, this will not work; but you did not say that either of these circumstances applied, so I have not addressed the issues.