I see no reason why it shouldn't be possible. Since the Raspberry Pi is a miniature of a processing unit, it should efficiently run multiple processes at the same time. In fact, just like most computers, the Pi is already running many background processes along with any process that you run.
And, speaking of downsides to running the two processes that you are talking about, there shouldn't be any unless both try accessing the same network port for their servers. But again, the Home-Automation.io doesn't run its own message broker by default, so there shouldn't be any problem as long as you don't subscribe using its MQTT broker.