I'm currently playing Hanon (1-20) at the target tempo as suggested by the book (108). My question is if there any reason to push myself playing it faster.
I can certainly raise the tempo if I put some effort to it, but I'm questioning the benefit of doing so. Should I push the temp with other stuff instead (Arpeggios / Scales) ?
Go as fast as you like IF you're sure your playing is clean and accurate. The technique will transfer to scales etc.
To determine if you have reasons to practice your Hanon exercises faster than the recommended tempo markings, it is helpful to consider your individual goals, and which aspects of your skills that you're trying to improve during a given practice session. Depending on your skill level, you might find that you could even double the recommended tempo, although that could introduce some bad habits or cause injury if you're not sufficiently skilled, and it may not be worth the risk to push yourself too far.
Some potential reasons for edging the tempo higher on your Hanon practice regimen might include:
If you choose to go faster, just be mindful about how and why you're doing it.
The other part of your question was whether you should instead push the tempo with scales and arpeggios, and I believe that the same principles apply. That being said, about 25 years ago, I was in the audience where Ellis Marsalis, Jr. was giving an amazing jazz concert, and he answered some questions afterwards. One person asked about whether he spent a lot of time practicing scales and arpeggios, and Ellis alluded that he got all the practice of those elements that he needed just by playing the music. I think that's a valuable guiding principle.
As long as your playing is completely clean I'd keep on going until you reach your limit. Exercises like Hanon will help your overall dexterity especially in classical pieces. But the main thing is to make sure that you are playing extremely and completely clean. No overlapping notes, no wrong notes at all. Otherwise this will not help at all.
Primarily, you should follow what Laurence said, which is to play clean and accurately.
To answer your question, I think that it is of course helpful to push the tempo a bit faster as some songs require it. For example, Beethoven's Moonlight Sonata - 3rd movement.
The benefit of pushing the tempo is that you will be able to play other pieces much better, as most pieces incoporate scales/appegios, so playing it faster will make playing scales fast much more comfortable as you're more familiar with it. Besides, there's only a certain number of scales on the piano, it's not like chess where there are an infinite number of positions. You'll definitely come familiar scales that (by that time) you'll have managed to play it at a high speed.
Tl; Dr
Try to push the tempo if you're playing clean and accurately at the current tempo you're at.
