Nothing, really. Your body is good at fighting that sort of thing, because it's literally happening every second of every day with billions of bacteria and environmental DNA. Even specialized organisms have a rather tough time getting by all the defenses.
For example, there are zero human-infecting raw-DNA pathogens.
If the goal is to use a bio computer to compromise a biological system, a more likely pathway would be to get the programmable system to construct something more like a prion (an auto-catalyzing protein complex) rather than faffing about with all the extra complexity and layers of indirection of an infectious virus.
If your target bio system is a plant rather than an animal, constructing a viroid might be worth the effort.
The biocomputing platform needs to not only assemble the sequence but also assemble the (or at least "a") protein encapsulation for initial delivery to an organism.
Hmmmm. If we're talking about actually wanting to infect someone with an engineered virus, then using something with sequences widely attributed to some other group could be used as cover.
Along the lines of "they were infected by a Russian variant of ebola". Or any other group, false flag operation type of thing.
Hopefully this doesn't become an actual thing. :(