If a circuit is connected to a separate power supply from an Arduino that communicates with it, should I connect the grounds together?
Your board outputs logic levels like +5V or 0V to control the LEDs, these are voltage, and voltage is relative, it is the difference in electric potential between two points. You can't have a voltage if you have only one wire, you need two. One of them is often called &quot;ground&quot; or &quot;0V&quot; and it is used as a voltage reference relative to which all other voltages are measured.
If you don't connect grounds, then the receiving chip on the other end has no way to know about that reference, so it will have no idea what voltage it is receiving. So it won't work.

can the connection to the data input on the Neopixels from the SBC potentially be dangerous in this scenario?

If you don't connect the grounds together then both devices will be at random potential relative to each other, which means that if current has to flow it will flow via what's connected, which is the IO pins of your chip, instead of just through copper. This is when it fries.
You would worry about connecting grounds if they were at different potentials, for example if you have long cables between devices, and these devices are earthed to different places in different buildings. Also if both devices are  earthed in a way that creates a long enough loop of Earth wire in the building that is closed by your ground connection, because when lightning strikes in the vicinity it will induce current into conductive loops. This is why Ethernet uses isolation transformers, for example.
But that would not be the case if you use two power supplies powered from the same socket. No worries in your case.