Does one need to wear tzitzis while exercising? Isn't it disrespectful to the tzitzis to make them all sweaty?
HaRav Binyomin Zilber zt"l wrote a responsum on this question to a rosh yeshiva in the US a few decades ago (see Az Nidberu 2, 55). He brings various Rishonim and for the most part says they should keep their tzitzis on, but doesn't discuss the issue of sweat. The question was posed in terms of discomfort on the basketball court. Although elsewhere HaRav Zilber writes forcefully against tucking tzitzis in, arguing that many people misunderstood the Arizal on this point, in the case of sports he says it's probably okay to tuck tzitzis in since it's temporary. But taking off the tallis katan altogether would be a chashash of beziyon mitzvah, chas vesholom.
First off: if you exercise at night it's a moot issue, as the mitzva of tzitzis is only in the daytime.
"Need" is a strong word. The strict obligation is to put fringes on any four-cornered garment I may happen to wear. No four-cornered garment, no obligation.
We tend to think of tzitzis as a special ritual garment (like tefilin), when in fact for thousands of years, it was just about taking your usual clothing (which, given Biblical fashion, was usually four-cornered) and putting fringes on it. So certainly it used to not be disrespectful to get them sweaty.
However, standard practice today is to opt to wear a four-cornered garment, fringes attached, when we can, to avoid being shirkers and to maximize our mitzva observance. "When we can" isn't clearly defined; if you chose to wear the garment for eight hours and not wear it for two, well, I'd say the cup is mostly full. (On the other hand, you could have easily had more of this mitzva, so why didn't you ...)
In gym class at my yeshiva, the most-devoted four or five guys had a dedicated "workout" pair of tzitzis that they wore (so they're not walking around the rest of the time in this sweaty garment). The rest of us just took off our tzitzis for gym, and put them back on afterwards.
I remember it being told (not so much taught) to me like this:
R' Moshe Feinstein held you should take them off out of respect.
R' JB Soloveitchik would remind people that they're washable (the implication being that he felt there was no need to take them off).
Source: Word of mouth when I was a teenager.
Prior to answering this, I would like to say the obvious- there is no hiyub to wear Tzitzit, only when one is wearing a four cornered garment (four corners or more). CYLOR before of course. I really recommend this SHIUR from Rav Mansour Shelit"a where he discusses something called Kelipat Noha (a Kabbalistic concepts). In summary it is about doing things LeShem Shamayim and its benefit. The reason I want to you to listen is because in there he says (paraphrasing) "There are two people, one who plays basketball with Sisit but not LESHEM Hashem, and the other who plays basketball without Sisit but for Hashem to be healthy for Hashem. The one who plays Leshem Shamaim is better." It is very interesting and highly recommended. Also there are some Posekim who might say it is disrespect to the Sisit to get them all sweaty. In my Yeshiva they didn't make the kids wear Tzitzit while playing sports, but a Kippa they enforced (which I disagree with totally- that's a totally different discussion).
Minhagim are by definition dependent on how the custom is accepted and practiced. For example, no one swims with a yarmulke. It would seem logical that one can go by the custom of his community. (Do diligent tzitzis wearers remove their tzitzis for sports)
It is a good idea that you have a special pair of tzitzis just for working a exercising. I wear a wool tallis katan everyday and a cotton one when I mow my yard or work in the garage. The reason for this is is because you are always supposed to see the tzitzis and remind yourself to do Hashem's mitzvos. I cannot imagine what might happen spiritually if one quit wearing tzitzis for even a minute! (Not saying you'll be struck by lightning, but...) Hope this helps. :)