Someone asked me for advice on their paper a few months ago. I pointed out an error. I received no reply. What should I do?

It's tempting to become emotionally involved here, but you need to resist that temptation. If the student (or their advisor) wanted to involve you more, they would have done so already. If would be courteous to write back, and if they remember to thank you in a final version, that would be nice, but discourtesy is not a big deal.

If you did, indeed, find a significant error in their work, it may have been an easy fix (and they are just not courteous) or it may have been a serious problem that could have undermined the entire result. They might be scrambling to fix it, might have even decided it is unfixable and moved on to something entirely different. Moreover, the situation you describe may have other non-scientific factors in play, as a student would typically not reach out unsolicited to an outside expert without in some way involving their advisor (even if just as a cc on the email). All of which is a long way of saying that you have no idea what might be going on with the student right now.

So, what about the other reasons you express for involving yourself?

• Should you be concerned about a manuscript with an error being publicly available on a preprint server? No, you should not. Preprint servers are filled with unreviewed work that is, indeed, likely to contain errors, and anybody accessing them should assume as much.
• Should you worry about an incorrect proof getting through peer review? No, you should not. They have your information, and if they choose to publish anyway and if then reviewers don't notice the problem, then that is not your responsibility. A correction or retraction would indeed be warranted and an unfortunately appropriate embarrassment to all involved.

All that said, if you are genuinely interested in making use of their results in your own work at this time, then you have an actual appropriate professional reason to contact them on that basis --- along with their advisor, who should be better versed in typical scientific practices. Say exactly your interest, and that you would like to know their progress for that reason. At that point, either an updated preprint or an offer for you to become more involved should be forthcoming --- and if not, I think it would be appropriate to consider yourself released to work in the area without constraint.

Do nothing. You've fulfilled your duty. The author needs to act, not you.

The author should be considering your advice. They may find you're wrong, or they may change their paper in response to your advice. Either way, they should have thanked you.

You shouldn't expect any changes to be released immediately. The author may be making many changes and they may want to complete all changes before releasing an update. Equally, they may be waiting for their co-authors. Assuming your advice helped, you should receive an acknowledgement.

It's possible that the author is unable to correct the paper alone. You could reach out to them, explain that you have a corrected proof, and offer to co-author the paper.

Before using the results in the future, you could reach out to the student, explain that you intend to use the results, but can't in the current form. Ask if they can update the preprint to contain a correct proof, perhaps even offering to co-author.

You seem convinced that you're right and they're wrong. Maybe you have a basis for that. You should entertain the possibility that you've falsely claimed that their proof is wrong. Perhaps you've missed something. Perhaps not. But always consider that possibility.