Can moving tables to another MySQL database help performance?
Generally, no. A few tangentially related things to note though:


You could use a symlink to move the /&lt;datadir&gt;/db2/ directory to another storage/block device. Or you could use the CREATE TABLE ... DATADIR=/../ option to move only the support_chat table to another storage/block device. That could make a big difference, assuming that you're frequently disk bound. 
With MySQL 5.6, using multiple databases/schemas can help to improve slave throughput (assuming you're using replication) because the parallelization was at the schema level:

Replication Slave Options and Variables --slave-parallel-workers

So with three databases/schemas, you would benefit from setting slave-parallel-workers=3.


For more info on the multi-threaded slave work in MySQL 5.6:

MySQL Replication High Performance: Multi-Threaded Slaves and
Group Commit (pdf)

Just FYI, the multi-threaded slave behavior has been greatly improved in MySQL 5.7 too, the biggest example being that you can now instead use (Lamport) logical clock based parallelization:

Replication Slave Options and Variables --slave-parallel-type