I've learned that Etudes are performance pieces, but what about piano studies like Hanon and Czerny?
Well pretty much anything can be played as a performance piece but many of them would be very dull to listen to. Personally I find Hanon extremely boring and I believe many other people do too. Czerny less so but still there is not enough in most of them to make then sufficiently interesting for the listener in a concert setting.
Having said that there is a huge list of studies - usually called Etudes - by great composers which work well as performance pieces. All of Chopin's and Liszt's Etudes as well as much of Scriabin, Kapustin, Busoni and many others.
So to directly address your question. You can play anything as a performance piece but you will not attract many listeners if the piece is not intended for that purpose. "Etude" and "Study" are often used interchangeably so that is not the distinction you should use. It comes down to the quality of the music and it's usually obvious if a piece is just a mechanical exercise or a well-constructed and interesting-to-listen-to piece of music.
Hope that helps.
I have taken a few works from Czerny, slowed them down and played them as prelude pieces in church. One day a woman commented how gorgeous a piece was that I played and I told her it was by Czerny. She immediately called a meeting of the liturgy committee and had a bylaw drawn up stating that the organist shall NOT PLAY FINGER EXERCISES for worship and they presented me with a list of acceptable composers. That next Sunday, I played several of Bach's two part inventions. Bach was an acceptable composer. I didn't have the heart (or stupidity) to tell them that the choir anthem that week was written by an atheist (Rutter).
I will be playing a concert on a theater organ this June and am deliberating playing a Czerny piece because it sounds really cool with the xylophone and bell stops.
The word etude means study. Most were composed for specific techniques formerly on harpsichord, then, when piano made its debut, for all the new techniques that were available. Czerny, Cramer, Bertini were prolific for the piano, Kreutzer and Rode for violin.
Chopin and Debussy took this writing to a different level, with their Etudes de Concert, and Liszt went even further, both in difficulty and Romantic quality. The latter ones, I guess would be ripe for performance. Although earlier ones were specifically for learning techniques, which is what studies are for, after all, any could be and have been used as performance pieces. It rather depends on the player and the audience.
Hanon and Czerny could well feature if one wanted to show particular facets of playing prowess to a selected audience - or could be segued into an interesting pastiche.
EDIT: not certain, but dynamics probably don't play a great part in studies. However, applying some of your own would perhaps make them much more like performances.
The short answer is yes, it can.
A better question is "Is this a good idea?", and only you can answer that. Try recording yourself playing the study and listen back to it, perhaps back-to-back with some other piano music you enjoy listening to. If you find it horribly dull, or much less enjoyable than the others, perhaps don't play it as part of a performance.
Definitely yes, but only if approached as a performance piece rather than as a study to begin with.
I feel compelled to explain what I mean by this. In theory, the above distinction should be redundant. We are musicians, and in theory, interpreting music should always be much more than simply reading notes from a sheet.
In practice, however, most people setting off to play a 'study' piece, approach it in a very different mindset to that of something that they would consider a 'musical' composition in its own right. People typically approach a study as a technical challenge, and feel satisfied that they 'mastered' the study when the technical element that it is supposed to teach (usually, but not always, one of a physical nature) has been 'mastered'. The result in musical effect is a piece that sounds technically challenging and possibly awe-inspiring, but ultimately clunky to listen to in the musical sense.
You will find that if you discard this mindset, and approach the piece from a musical perspective, all but the most boring studies (and sometimes even those!) suddenly take on a life of their own, and become very musical and interesting pieces to perform, with lots of exciting stuff happening beyond the purely technical stuff. Czerny was one of my favourite composers to perform back when I used to be a pianist, and I always got great feedback on the melodies and interesting musical patterns involved, because I always tried to make that my focus and consider the piece as a whole, rather than a the sum of its 'technical' aspects, serving only as a 'show-off' exercise. There's some really nice gems in Czenrny's etudes that really bring out beautiful music when approached with this mindset, but could easily be reduced to a salad of notes if approached only for their technical challenge.
I feel that, perhaps the reason Chopin's etudes traditionally tend to be considered as more 'performance' pieces than Czerny's, is a) because Chopin has earned the reputation of having more 'musical' pieces, such that even his etudes tend to be approached from the musical standpoint to begin with, thus more easily achieving a performance mindset and quality, and b) because some etudes aren't actually that technically challenging to begin with (at least not in the physical sense -- many of Chopin's etudes are exercises in producing a specific musical effect, such as the standing out of an inner melody, rather than technical exercises in speed or finger stretch).
Finally, just to get a bit more provocative here, I will directly contradict the (currently) accepted answer as regards Hanon. If you simply approach it as practicing slightly more elaborate scales, then of course there's no performance value there. But this goes directly against what it means to be a musician. A good pianist / musician can find musical meaning and bring about beauty, even in a Hanon study. It's all about the interpretation, and how you approach a piece. Alas, part of the bigger problem here is that even professional teachers and performers tend to approach these simply as 'finger strength' exercises, so there's not that many positive examples out in the wild to be inspired from. But if you make a point of trying to bring out the music even in something as simple and structured as a Hanon study, it will serve you a lot better in the long run than if you approached them purely as 'finger stretching exercises'.
There is no rule for what should or should not make it into a performance. If your audience begins to leave the room or fall asleep, you may have gone wrong somewhere, and I am not sure that's even true.
I have given on one occasion a recital strictly made of Etudes & Study pieces. In that concert, the only reason why I played some Czerny & Hanon pieces was to help listeners appreciate the genius of other composers like Chopin, who could write study pieces that were also masterpieces as opposed to sounding "just" like studies.
Note also that Liszt wrote some Etudes named "Concert Etudes" where the explicit intent was for them to be performance pieces. In fact, these are more about performance than studies, unlike the work of Chopin, who focused very much on one or a few specific technical challenges in each of his Etudes. Making the beauty comes out of an Etude is what makes his body of work on Etudes unsurpassed in my opinion.