I've read tons and seen videos about testing sets by touching this or wiggling that but I've never read or seen anything on having an antenna and ground connected while the tests were being done.

I've heard that if the output doesn't have some kind of load it could fry something.

How important is it to hook them up or do they have to be there at all?

An antenna and ground would be a good idea if doing a functional test of a sets reception. Many sets will have no audio output without some kind of antenna, and less often, a ground.

If you are only measuring power supply voltages or voltages at tube sockets, then an antenna and ground are not required.

For many sets, a speaker is required to be hooked up for the set to function at all. Speaker field coils serve as an important part of the radios power supply in many older sets.

Prolonged operation without a speaker is typically not recommended.
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My response above concerning speakers is directed at your concern about "output connection".
If you were only talking about the antenna, then that would technically be an input.
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:My response above concerning speakers is directed at your concern about "output connection".
:If you were only talking about the antenna, then that would technically be an input.
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Lewis