I think so far objective answers are lacking, as all of them seem to assume this notion of "voting is a right" and presume that the electorate should include the largest number of people. What is the basis for that assumption?
I will expand on some of my comments on an attempt of a systematic answer, and I should note that while I am not sure what the scope of the question was, I do focus on the USA. I am not a U.S. American myself, but I have a stake there in the USA and I believe the USA was the best example of a Republic the world has ever known. Even though it has dramatically changed in the last decade and continues to change at breakneck speed.
So, what is the basis for that assumption that the electorate should include the largest number of people in some form and that some test is categorically not a good idea?
Someone above said that even age is already a test. As soon as you have criteria for inclusion, you have a test. Why don't we let babies vote as soon as they can hold a pencil? As soon as they can read? Why don't we explain the implication of the elections to 10 year old children in a way they can understand as then poll them for their opinion? It's their future after all! These arguments are being made, they have been made when the age was lowered to now 18 years and new proposals to lower to 16 years have been made in earnest. But 16 is just a number. 10 year old kids are now participating in climate protests, so why not ask them to vote? Why not 8 year olds? I heard often that TV content is targeted to people with cognitive abilities of 8 year old children (cannot find the source for it now), so why not 8 year olds? It's their future!
What is the test? There is a cognitive ability assumption on the age criteria. But there is also an independence argument. If you allowed children of 4 years to vote, chances are that they would be almost 100% guided by their parents or random error, and that would not be much better with 8 years. The key factor is that the children are dependents. They lack the experience of self-responsibility. Now consider self-responsibility as a criteria, one that can be objectively measured while not being a "test" at all. A 21 year old college student supported by their parents is also dependent. And so is a 30 year old welfare recipient. And why would you not call a government employee or contractor a dependent?
These are serious questions.
The assumption of universal (and ever expanding) franchise is also not based in the formation of the USA, where the franchise was much more restricted. Even leaving out the nasty mess of slavery and the reconstruction era and beyond, the franchise had from the beginning been restricted to land owners, wasn't it? And who was a land owner? It wasn't hard to be a land owner if you were a settler in the early USA, as people would become land owners by homesteading, it was accessible to most who came, except those who came under a contract where they committed themselves to a certain amount of time of labor for subsistence only; after which time was up, they could go and homestead their own patch near the frontier. The principle of this was that it gave the franchise to the people who have a stake in the country beyond their immediate personal lives.
The restriction of voting to land owners has often been framed as Americas "oppressive" and "shameful" past. But it isn't honest to simply impute an oppressive motive in the formulation of the USA's system of voting and government as it originally was. A far more realistic assessment should at least very seriously consider the issue of dependence. This dependence might very well turn into a license to vote. You might have to show that you have been living independently of others, supporting yourself from your own economic activity, and not receiving your livelihood from government (regardless if a politician, a government employee, grantee, or contractor, or welfare recipient.)
The great Civil War was a major push to extend the franchise, by considering service in the military during the war (and the many that came after) a ground for the voting rights. Still that is a notion of voting right in exchange for some responsibility for the country. But it went on from there, quickly, to universal franchise, suffragettes and all. Note that the principal argument against the suffragette movement also has not been some ominous "oppression" motive, but the fact that most women were de-facto dependents of their husbands.
However, the notion of "license to vote" is probably usually seen as some kind of aptitude test. And the big problem is that this is very open to manipulation by technocrats. Such technocracy is always firmly in the hand of left statist politicians. You can even see this in the answer that brought up the term: "Technocracies (governments led by experts)", he says, "can often [govern] better." This is a prejudice that never dies, despite technocracies always failing. The idea of technocratic planning government arose in the USA in the Progressive Era, and most of their ideas failed or were taken to murderous extremes elsewhere (e.g., Hitler learning his share of race ideology from the U.S. Progressive Era leaders.) Eugenics, forced sterilization programs and other excesses like the Tuskegee Experiment are one thing, but the simple fact remains that planned economies always fail to fare better than free market economy.
So the issue with such aptitude tests is that they can be easily used to measure the voters agreement with big government ideologies and hence benefiting the politicians and the bureaucrat-military-industrial-academic complex that feeds off entirely from their government dealings. If you stand on the left, you would probably be worried about the tests to be used for the ever luring racial discrimination in all its ever elusive subtleties and dog whistles and such, but if you are on the right you should at least be equally concerned, as by now you need no imagination to see how tests might be deployed to weed out deplorable low life of flyover country to interfere in the grand plans for a better world that the coastal elites have made for all of us.
So, this, I would say is the biggest reason not to want aptitude tests.
But to go back to the earlier thoughts, we should be aware that we always make some test that measures eligibility to vote on some criteria. Age, criminal record of felonies, and not to forget citizenship. And all of these criteria are being disputed and adjusted, striving for ever more inclusion. However, this is at odds with the original designs of the voting and governing system, among some notable others. Especially the 17th amendment of 1917 which, some say, ended the US Republic by doing away with the control of the states over the federal government. Already you can hear lots of people chiseling away on the Electoral College, chasing after the primacy of the "popular vote", and at that rate it is only a matter of time until the Senate will be re-structured to strive to become more demographically representative. This changes America completely.
And all of this is not a new observation at all, as Alexis de Toqueville already warned about even before the Civil War era.
Tocqueville warned that modern democracy may be adept at inventing new
forms of tyranny because radical equality could lead to the
materialism of an expanding bourgeoisie and to the selfishness of
individualism. "In such conditions, we might become so enamored with
'a relaxed love of present enjoyments' that we lose interest in the
future of our descendants...and meekly allow ourselves to be led in
ignorance by a despotic force all the more powerful because it does
not resemble one", wrote The New Yorker's James Wood. Tocqueville
worried that if despotism were to take root in a modern democracy, it
would be a much more dangerous version than the oppression under the
Roman emperors or tyrants of the past who could only exert a
pernicious influence on a small group of people at a time. 
The issue comes back to that of responsibility and independence, and having a stake in the future of the country, not just one's personal present well-being. Those who are not dependents but are responsible for dependents (wife, children, elderly parents) and those who own property, are much more prone to consider future value over instant gratification. That this is a matter of actual shouldering of responsibility rather than some "wisdom of age" is clear very quickly when you consider retirees as a voting block which politicians frequently have to pander to with promises to not dismantle retirement schemes and not lower pensions to protect the fund for the future generations. Wisdom does not come with age, but with economical independence, and ends with dependence, whenever it sets in, even after youth and even before the old age, primarily as a beneficiary of any government programs.
This principle of responsibility as the primary measure of eligibility to vote has been disregarded in the chase for the "popular vote" and is likely to haunt America for some time to come.
It has been argued that democracy with a large electoral base leads to more stable, peaceful people(s) and fewer revolutions. This idea, however, is hardly proven by a longer historical horizon, as Hitler and Mussolini were democratically elected, Hugo Chavez in Venezuela was democratically elected, and the Civil War in Spain in the 1930s also ended a period of democracy with large voter base. Rather the experience with these failures of democracy are a confirmation of de Toqueville's warnings, and also those of Friedrich A. Hayek in "The Road to Serfdom".
Very briefly my answer can be summarized:
- Aptitude tests are political filters and the left usually rejects them as racist but the conservatives/libertarians have even more reason to reject them as empowering tyranny.
- There is always some test and there are movements to abandon all limitations on the electorate, these movements, however, do not want to consider the importance of self-reliance and responsibility vs. dependence, as they can manipulate dependent voters.
- The original Republic of the USA considered limited the franchise on a measure of responsibility, and only slowly it was expanded in exchange for service to the central government (Civil War effort) and later the idea of this "voting as a human right", and a subsequent revision of the original ideas of responsibility to suppose a negative motive of oppression.
- It has been argued that democracy with a large electoral base leads to more stable, peaceful people(s) and fewer revolutions. This idea, however, is hardly proven by a longer historical horizon, as Hitler and Mussolini were democratically elected, Hugo Chavez in Venezuela was democratically elected, and the Civil War in Spain in the 1930s also ended a period of democracy with large voter base.