Does adjusting a domain name's TTL actually matter?

I'm aware that tomorrow the public IP address of one of our production servers is going to be changed.

The TTL on that A record is currently set to 3 hours.

Will adjusting the TTL on that A record to something lower like 1 minute actually work (the domain registrar does allow specifying minutes!), so that users DNS will only be pointing to the old server for a maximum of 1 minute after we switch that A record to the new public IP address?

They're not supposed to, but some DNS services may treat this as more of a suggestion than a hard rule. They may honor the setting down to some minimum, or they may ignore your TTL completely and always use their own setting (I've heard that 2 days is, or at least was, common). You need to be aware there is nothing you can do that will make those providers update any faster, and therefore some requests will end up going to the old address for some time after you make the change.

Ideally in this case, you want to cut over to a new IP address while you still have some control of the old address, such that your server can be set to handle requests via both addresses for a small interim period.

Additionally, some DNS services charge you per request (or per million requests). Moving from 3 hours to 1 minute will increase your DNS requests by a factor 180... you'll get 180 times as many requests as before. It's not likely to break the bank, but just make sure you're prepared for that.

As an example, I have DNS service for a rather small web site where I spend about $20 per year for them to service 5 million requests per month. I admit that I'm not actually sure whether they'll just bill me or stop handling requests if I ever exceed that, though I expect it's the former. Right now I tend to only get about 1/2 million requests per month, but I wonder what would happen if I changed my TTL setting to get 180 times as many more and left it that way for too long.

Still, most DNS services will honor your 1 minute setting. This will help smooth the changeover to the new address, and it's not likely to hurt you at all as long you're careful. Just remember to do this at least 3 hours (the old TTL) ahead of the change. There's no point doing it much earlier; any provider that would need to see the change sooner is not honoring the setting anyway. And, of course, don't forget to put it back when you're done.

You may also want to reference this question:

Migrating DNS Providers

It's a bit different than yours, but some of the issues involved are similar.

Only if you make that change to the TTL more than three hours in advance of the IP address change.

Remember that the TTL tells other DNS servers how long to cache records. So you must reduce it at least that long in advance of your desired change.