Competing conditioned/combustion air flow; worry about CO?
You're right to worry that opening up that room to the rest of the basement will mix combustion gases with the rest of the air.

I'm guessing from some clues (oil-fired boiler, baseboard heat) that you're in a cooler climate.

You could cool that space separately from the rest of the house.  Can you draw in outside air on a thermostat-triggered vent fan?  A/V and network equipment can operate at a higher temperature than we generally keep our homes.  85 degrees generally isn't a problem if there's enough circulation to prevent hot spots.

If it were my house, I would build or install a cabinet to hold the equipment.  Use a short piece of flex duct to draw in cool outside air at the bottom.  Use another with a thermostatically controlled fan to exhaust it at the top.  On the exterior, you would use something like a dryer vent hood to prevent water, bugs, and rodents from getting in.  Inside the cabinet, mount a couple additional fans to ensure everything gets adequate circulation.

If the risks of outside dust and temperature swings are concerning, you could do the same solution and vent it into the basement.  If you did that, you'd need to make sure the cabinet and all ductwork are sealed tightly (again with the combustion gases).  You'd be able to skip the thermostat on the fan as well.  But you'd have to listen to that fan whine pretty much all the time you're in the basement.  A fan that can move adequate air will make some noise.