There are two mikrotik routers 192.168.0.2 and 192.168.0.3 on my Local network. I need that client computers which use gateway 192.168.0.2 would use whole internet from that gateway except for port 6667 whose all traffic to internet should go through 192.168.0.3.
What rule should I type? Help please

I would try dst-nat rule on traffic destined for port 6667 arriving at 192.168.0.2, redirecting it to 192.168.0.3 port 6667. You can also drop all traffic destined for ports other than 6667 on 192.168.0.3.

However, I am not sure if MT can redirect traffic to a different host on the same network.

It can't!
Try giving 192.168.1.1...

Cheers...