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I'm writing a fantasy story where magic is common and wizards follow a strict code of conduct. One of those tenets is "never stray from humanity and do not seek immortality or godhood." While I admit that becoming a god might be a complicated thing to restrict from literal reality warpers, how would governments go about policing and restricting people from obtaining eternal life through magical (and perhaps technological) ways? One other question I have for this is, how would a ban on immortality impact healing magic and medicine if it could potentially be used to prolong life? What other kinds of impacts on society would take place if immortality was illegal?

A strict maximum age.

Genesis 6:3:

And the LORD said: 'My spirit shall not abide in man for ever, for that he also is flesh; therefore shall his days be a hundred and twenty years.'

You could implement this sentiment into law. People who reach 120 years are required to die. If they don't agree to medically-assisted suicide, they are executed.

This is of course a rather harsh policy, but some worlds are harsh so I don't limit my worldbuilding to only pleasant ideas.

This policy would stop your magically immortal people from living longer than 120 years if you kept good enough records of when people were born and could find the violators. As far as I know, false-positives would be extrememly rare. If this policy had been in effect globally here on Earth, then in all of recorded history it would have only killed one mortal person who was still alive by non-magical means: Jean Calment and she would have been required to die two years, 164 days before her natural death. If you extended the limit to 125 years, then no non-magical old person would have been killed.

I made a joke in earlier comments, but then got to thinking further... does your magic allow for resurrection? If so, you could have everyone required to come in every 10 years, say, as part of getting drivers license renewed, and a government official shoots you in the head. Or injects you with poison. Or something. If you die, it proves you haven’t crossed the line, so they resurrect you. If you don’t die, they try twice more to kill you, and if that fails, you get ejected into space/flung into prison/removed from history by a time cop who prevents your conception/etc.

The most powerful institutions are also the ones who most excel at tracking your income and taxes.

Just make it so that your income taxes are adjusted by age past a certain limit - say, after 100 any retirement plans are cancelled and your taxes on income are doubled every decade.

This would in effect cause a maximum age to be enforced by law.

Of course, the most rich and powerful could always evade taxes and move their wealth to havens, or deal in manaCoin. But then again, fantasy is no fun without a powerful lich for the protagonists to lynch.

Other wizards need to take care of this problem.

If a magic user has skills enough to achieve immortality, he or she has skills enough to dupe / buy off / intimidate the cops. The regular civilian police are going to be hopeless against some necromancer lich. Your only hope is other wizards.

1: Other wizards will probably know who among them is up to immortality shenanigans. Wizards have scrying glasses and divinations and such.

2: Nonimmortal wizards will likely be envious and seek to prevent that sort of cheating.

3: Assembled other wizards might have the skills to actually take on an immortal. I could imagine a "Witcher" sort of magic user who made this a regular gig.

4: These sorts of risky civic-minded actions will put the wizards willing to undertake them in a good light with the authorities, who might grant such wizards privileges and resources otherwise unavailable.

Theft and murder are against the law, yet it happens every day. Even though immortality is illegal, not matter what you do to enforce it, someone will always find a way to get away with it.

Detailed birth records, magical observations and age tests and bureaucratic oversight are all nice and all, but it will only catch the honest.

One method that may help catch most, although draconic, a randomized culling. If you have death squads that go out and wipe out 2-5% of the population each year, odds are you will have wiped out ~99.73% within 100 years. Sorry, I'm a little fuzzy on statistics, but if you account for normal birth/death rate and take the statistics out to the 6th standard deviation, then you should ensure most of the population gets causght up in your death squad sweeps.

Is ontological immortality possible (i.e., one cannot die in any way, and will always regenerate from any trauma, like the Greek gods), or merely functional immortality (you won't die naturally, but can still be killed)?

If the former, prison is always an option.

If the latter, just impose the death penalty (or merely eternal prison) for anyone who is found to have become functionally immortal. If you want to allow the imposition of functional immortality as a last-ditch effort for treating certain injuries and diseases, then put a maximum lifespan cap on it: if you can no longer die by natural means, the government will put you to death on X date, and you agree to that as a condition of treatment.

How this would impact healing magic and medicine depends on the legal definition of functional immortality. Does it count if you have to keep having repeated rejuvenation treatments, such that your immortality isn't actually inherent to you but rather is a feature of your living conditions, or does it only count if you make yourself permanently immortal? If the intended effect is just to ensure that there is nobody over a certain maximum age, then none of that matters; you just impose the death penalty for anyone who can be proven to have past that age (like in Logan's Run), no matter what. Just make it high enough that no one would ever naturally live that long (unlike Logan's Run). But if the intent is merely to prevent people from becoming superhuman, then there need be no particular effect on healing magic or medicine, and no need to impose a lifespan cap.

The maximum age limit that others have suggested is a good start, but can still be circumvented with some forged documents to fake your age. Also, doesn’t do much about godhood.

Think about how government and law enforcement try to control contraband like explosives: they limit the legal availability of the materials known (or suspected) to be used for immortality and omnipotence. Thaddeus the Green has been buying up all the quicksilver in the tri-county area? The feds have a judge give them a warrant to scry his laboratory. There would probably be whole teams of wizards with license to do theoretical research into immortality to figure out the probable methods, so long as they share what they find with the government (and submit to some extra transparency scrutiny, just in case).