Suppose X is a citizen of two countries, A and B, both of which permit dual nationality. Should they obtain both passports and renew both of them when about to expire?
If X can possibly afford the fees, it is much better to hold actual passports for both nationalities.
A country may require the use of its passport by its citizens when entering or leaving the country. The US is an example, although the practical penalty appears to be a delay and lecture.
On the other hand, some countries require the mandatory use of its passport by its citizens when at least entering the country. e.g. X wants to visit country A (and doesn't have country A's passport) and country A has visa requirements for country B citizens. In such a case X would need to obtain a country A visa on X's country B passport prior to travel. A real-world example is a dual US/PAK citizen traveling to PAK having just a US passport.
X may have a need to travel urgently to a country that permits visa-free travel for one of the passports, but not the other. The delay in either obtaining a visa or obtaining a passport after finding out about the need to travel may be inconvenient. In general, you cannot expect other countries to allow the travel benefits of being a citizen of country A unless you can present a country A passport.
Generally, not holding a passport does not exempt you from the obligations of a citizenship. For example, the US tax obligations of a US citizen are not affected by whether the citizen has, or has ever had, a US passport.
This hugely depends on the two countries involved. For example, if one of the countries requires you to enter and leave with their passport (like the US does) then it's a good idea to keep that passport up to date if you intend to visit that country. On the other hand, if both of the countries passports are accepted in most places, such as two European countries, then there isn't much benefit to holding two.
I have dual citizenship and stopped renewing one of them years ago. But a change of residence would make the one I dropped valuable, and I would switch.
The big advantage to having both of your passports is that you can enforce your right of entry. As a citizen of a country, you have an unassailable right to enter that country. A non-citizen can be refused entry. If you don't present that country's passport when you enter it, you technically don't have the absolute right to enter, although many countries will give you the right if they can be satisfied that you are actually a citizen of theirs. A passport is pretty compelling proof of citizenship (obscure non-citizen passport situations excepted).
As mentioned by other posters, one passport may give you visa-free access to countries that the others don't, so you may prefer to use one passport over the other in some circumstances. On the other hand, if both your passports are EU Schengen-area passports, for example, there may be little advantage to having both, as you have right of entry into the Schengen area with either one.
I know that there are some countries where failing to renew a passport under certain conditions means that you are no longer interested in citizenship, and citizenship will automatically be revoked.
For example, this is the case for citizens of the Netherlands who have dual citizenship and have not lived in the Netherlands for the past ten years. If they fail to renew their passport on time, their citizenship is automatically revoked.
I also recall hearing of situations where not renewing a passport will make life really difficult if you want to get that passport later on. I don't remember any specifics though.
I recommend checking citizenship and passport rules to ensure that not renewing a passport won't have any unanticipated consequences.
Many people live in one country and never leave it, even when they have citizenship in an other country. In that case it is very likely they do not want to have a passport for the country they do not live. In fact, many people do not even have one passport.
And in some cases it might be prudent not to be able to prove citizenship, like when you may be forced to join their army, even when you do not speak the language and have no ties (other than one of your parents having been born there) to the country.
Not having a passport may not be enough to avoid military service, but not having applied for a passport might make a difference in some cases.
Not unless they've a need.
There are certainly countries it's easier to get into with a passport for one of my citizenships than the other, and vice versa. Not having a pressing need to visit any of them, why would I waste money on a passport I'm not going to use?
If I was planning to visit a country that is more welcoming of the citizenship I don't have a passport for, I would certainly consider getting it (though probably not; the few places that would prefer it aren't particularly unwelcoming of my other citizenship either, but for other citizenships it can make a massive difference).
If I was expecting to have some need to prove that other citizenship for some reason, I'd probably get a passport.
If I was expecting to have to leave the country in a hurry (not a bad thing to assume as a possibility wherever one was) and the citizenships had very different treatments in many countries (in my case they do not), I'd consider having both.
I'm not going to spend time and money on a piece of paper that'll just get itself lost somewhere in my office though.
Just to give one concrete example of the potential benefits- a colleague has three passports (issued by three different countries, on three different continents). For visiting a certain country (Turkey) the cost of a visa is $60 US, $20 US or $0 US, depending on which of their passports they use. If they tried it with a US passport they would not be granted an e-visa at all.
The passport is usually just proof of a condition that already exists, so in most cases one would renew it if the document might be useful to you.
There are however possibly circumstances in which renewing or not renewing a passport conveys intent. This is more likely to come up with regard to refugee status and that sort of thing. In which case a qualified professional had better review the case with reference to the individual situation and countries involved.