I am permitted to mix not open-source code with GPL code on my computer if I don't distribute it.
Is deploying it to a VM on a remote server considered distribution?
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Copyright law cares about legal entities, not computers.
Unless AGPL is involved:
It doesn't matter how many different real or virtual computers are involved -- the question is, did you make it available to a different legal entity -- did you meet the legal definition of 'publication'. See the actual definition.
Now, for AGPL, the question will be whether you created a service open to the public -- again, completely independently of how many computers, real or virtual, were involved.